Turns out winning in 5 turns is possible.
Strat:
Rounds 1&2: Add 2 level 2 skeletons and anything else OR 1 level 2 skeleton and 1 level 2 skull. Best is 2 level 2 skeletons and archer/stone thrower. A level 2 archer is good for rounds 2-3 when you don't spy since it can target any lane. Always get skeletons first if they're there so you can afford everything. You can afford having some inefficiency, with a 4 castle damage combo instead of the ideal 5 damage for one or both of these rounds if you get lucky later.
Round 1:
Round 2:
Rounds 3-5: You want summon spells, rats and more level 2 skeletons (with the aim of getting at least one level 3 summon book by round 4/5). Bones are good if you already have skulls, but you probably don't want to make a new skull unless it's the best option available. You can probably afford 1/2 other buys if you must and they can protect the castle damage dealers. Use the summon spells as fodder to protect the level two skeletons/skulls as they do more damage to the castle. Rats should be protected the most as they do the most damage to the castle vs their ability to take out units. Level two rats are better than two level 1 rats, but level 3 isn't useful until fighting the king. When you get the level 3 summon book, potentially buy the tier 3 unit produced (ideally it's a summoner). For rounds 4-5, use the spy as you may need to sacrifice a lane to deal more damage. Don't merge summoners into level 2.
Round 3:
Round 5:
King Fight: 3 of any: summoner, bard, rocker, rats, bat spell to remove shield. Use spy to make sure you don't die and pick the best options. Don't bother with princes or knights in other lanes.
(Note that I altered the market probabilities to reduce the number of retries I needed when I was testing out the strategy, but it is still doable if you get extremely lucky. Using this strat across 6 rounds is much more likely and gives some leeway in terms of luck.)
I think five rounds is the best possible as the only way (I think) to do 60 castle damage in 4 rounds even if no units die is:
Round 1: 2 level 2 skeletons + 1 level 1 skull = 5 damage
Round 2 : 4 level 2 skeletons + 1 level 2 skull = 11 damage
Round 3 : 1 level 2 rat + 1 level 1 rat + 4 level 2 skeletons + 1 level 2 skull = 18 damage
Round 4 : 3 level 2 rats + 4 level 2 skeletons + 1 level 2 skull = 26 damage
Total : 5 + 11 + 18 + 26 = 60 exactly, but I can't think of anyway for all units to survive every round without losing more than 9 health.
