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All of these spells are in D&D 5e; which is where Temptation's Ballad gets its spells from as well

You can find Counterspell in the Player's Handbook, Phantasmal Force in the Player's Handbook, Dissonant Whispers in the Player's Handbook, and Synaptic Static in Xanathar's Guide to Everything, but what each of these spells do in D&D 5e can also be found in 5e spell list apps and sites.

Counterspell, Phantasmal Force, and Dissonant Whispers are all spells that came from older editions and were just updated for D&D 5e. Counterspell allows a caster to snuff out a spell as it is being cast but Bards do not naturally learn it unless it is one of their Magical Secrets (a trick only bards get that lets them learn any spell in existence regardless of class, species, age, etc; it lets them ignore any and every requirement for learning a spell outside of needing to be a high enough level of caster to cast it i.e.: a 9th level spell such as wish requires a Bard to be able to cast 9th level spells for their magical secrets to let them learn it, EDIT: I forgot to mention the first time around but Bards get 6 spells through their magical secrets with one specialization (Lore Bard) getting 2 additional Magical secrets much earlier on than any other bard specialization for a total of 8 if a Bard goes down that one specific path), Phantasmal Force creates a Phantasm that the target perceives and will believe is real that can interact with them but only they can see it (really versatile and whack spell that is as effective as the caster is clever), and Dissonant Whispers via methods of fear, compulsion, or the like psychically damages the target and forces them to run away from the caster as far as they can without doing something like falling into a pit (knowingly; it can force them to fall into a hidden pit); casters can use this for many things but Bards will usually do it to buy distance from a target to flee or reposition, or to open a target up to attacks of opportunity from their allies since nobody is immune to the effect even if they are immune to psychic damage, but again the more clever the caster the more busted it is. Kinda how it is for like all of the Bard's kit; perfect at support and perfect at manipulation and control, though Shirokoi can adjust everything to whatever extents he wishes cause ultimately it is his world, game, and story; it just uses the D&D ruleset (which the 5E rulebooks themselves say can be tweaked and broken  in any way the DM wishes, and in Temptation's Ballad, Shirokoi is the DM).

Synaptic Static is Fireball if Fireball did psychic damage (hurt your psyche and mind to the point of actually damaging you), and then just makes the person worse at everything for a minute without requiring concentration, but it also notably (and directly) states that it also penalizes concentration checks for those who fall under its effects; so it is a pretty strong anti-caster spell. It penalizes everything by muddling the thoughts of its targets, and if I was to say anything inspired it I would say it was some designer somewhere saying "What if we took fireball, made it do Psychic damage which almost nothing resists, made it require a target overcoming its effects with sheer intellect (which is the usual stat for any character to have suck), and then make it so everyone who fails to overcome it just gets worse at everything for a bit".

If you meant the inspiration behind saying these spells fit Cole though; Counterspell is the perfect anti-caster spell and since it is an instant reaction spell it would still be easy enough for him to cast it in a subtle enough to hide way like he does prestidigitation, be it a specific strum of the lute or the like, but really cause even though every bard is recommended to take it in actual D&D as a magical secret (because only two specific kinds of specialized casters are better at it than bards are for reasons that can be looked up for anyone who is curious, one of them being a specific kind of wizard subclass/specialization (abjuration) and the other (and best one) being the lore subclass/specialization of Bard, and other than those two specific specializations every caster will be worse at it than a Bard) Cole notably enjoys f@qing with and toying with people, and the easiest way for Cole to deny any Noble wizard from accomplishing anything would just be to Nope their magic away. Of course Bards have to get pretty strong before Magical Secrets (outside of Lore Bards who get it really early at 6th level out of 20 levels and then get a bunch of them later), all but one kind of Bard needing to reach 10th level (again out of 20 levels) to get access to their first few magical secrets, so Cole probably wont be playing around with that until he starts to actually accept his spellcasting more openly (unless he ends up becoming a Lore Bard, but I don't even know if he has specialized yet and I don't think his abilities would go lore over options like Eloquence (the manipulative and charismatic bard) or the like for early secrets, no idea though).

Dissonant Whispers is a "Get-out-of-Melee-Range-Jail-Free" spell and when mixed with his illusions and usual habits of playing tricks could come in a lot of handy; such as when someone is trying to grab him and he prefers not to get strangled or mauled at that current moment, or when he wants to force someone to set off a hidden trap; run into the "loving arms" of an angry monster, cower back into the arms of someone else to humiliate them, the possibilities are endless depending on the scenario and setup.

Phantasmal Force lets him F@q with someone's head; since they will believe it as real, it makes all sounds and noises, and even though they will pass through it (such as falling through a Phantasmal Bridge) if they try to touch it; they will still believe it was real no matter how they have to rationalize it. Ya want someone to appear mad in broad daylight? Phantasmal force since only the target can see it, and if it stabs them or does anything to them it will be real enough to them that they will still take damage. Ya want someone to attack a friend? Phantasm yourself over them; they will believe the lies it speaks as it stands in the same place as their friend; the fact that they overlap makes rationalizing the person having been disguised as their friend easy. Hell; making a specific person see a second Cole or second Sidd on its own could really mess with a target; since they will believe the Phantasm is real, and since the Phantasm can be anything that can fit within the space of a 10foot by 10foot cube be it living or inanimate; it can do or be about anything Cole can imagine. 2nd level spell too (counterspell is 3rd, dissonant whispers is 1st level, Synaptic static is 5th level) so Cole is probably just about to hit the threshold where he can learn it (Bards like all Full Casters get 2nd Level spells when they hit 3rd level in their class; 3rd level is also where Bards Specialize so Cole if he is a low level Bard (which would make sense) may be unable to hit that threshold until he is ready to be real with Marrow about his casting, or he is just about to hit that point, or he is going to have to train with Sidd's mom more, but he also could already be past that point, time will tell and I can only remember Cole casting 1st level spells and 0th-level/Cantrips up to this point (I forget if he has cast suggestion or not but I can 100% guarantee without any doubt he will be slinging that spell around the moment he has it and it is a 2nd level spell which he gets access to as soon as he is a 3rd level bard), but Disguise Self is 1st Level which Bards can cast those at start, Charm person is 1st level, Prestidigitation is a Cantrip/0th level. all spellcasters can cast level 1 and 0 spells upon the manifestation of their magical abilities, just they can only cast level 1 spells 2 times per long rest as a level 1 caster and 3 times as a level 2 caster and level 0 spells are always infinite, that and Cole casting Disguise Self twice in a row and getting super drained by it also implies he would be either a 1st level Bard and that is the full extent of his casting without a rest or that he is a 2nd level Bard and only has another Charm Person or Disguise Self in his tank before he is stuck with Mage Hand and Prestidigitation, but even then if he was a 2nd level Bard he would probably be learning another spell soon. Sorry; I am a colossal nerd, and once again; Shirokoi gets to write and work with it all however he wants and that would genuinely still be rules as written D&D 5e since it again states that the final say on every rule and everything is the DM's and they can rewrite any rule they wish.

Synaptic static is cause getting a large group of people to suddenly become worse at everything seems up Cole's alley, but also cause I want to see a much stronger Cole hit Calder with it and ruin his entire career again later in the story.

Sorry about the huge walls of text. I just wanted to Geek out a little and speculate a little; thank you for your time!