I don't think SP regen should ever be zero, as it would cause some issues. In the past, there was a problem with the power swing of having 1 compared to 0 SP, forcing everyone to get dex ASAP. This severely limited early build possibilities and was generally just no good. SP not regenerating as a base would cause the same thing to happen again.
An alternative to reduce the time stop effect would be to limit the amount of SP movement you can use to a few per turn, somewhere around 3? This would still give the movement abilities like sprint a purpose, or they could even be changed to increase the cap. That would fix the problem without messing with the balance of duelist, barb with charge, or creating issues where SP can regenerate too fast and be way too powerful. It also guarantees to remove the time stopping issues, without having to remove a lot of SP related items. If the SP items remain, players will either be able to stack SP, or if they are changed to regen, get insane SP regen.
Another important thing about allowing maxSP increasing items to remain, is they are very important for giving more flexibility to builds. If SP items didn't exist, a lot of builds would be forced into more dex. Talking about dex, if it gave increased SP regen this would limit dex based builds a lot. Getting SP back faster is just so strong, there would be few viable options apart from to fully invest in dex. Any changes really need to consider this, as the balance of RF4 attributes is pretty good right now. After your talents are unlocked, you can take most attributes and not be severely handicapping yourself, which opens up a lot of possibilities and is fun.
This would not do anything about the overall SP regen rate, but I would need to understand why you feel that SP regen out of battle feels bad first to have ideas on that. What part of resting for SP feels unreasonable? Does it use too much food, or something else? It feels perfectly natural to me that after a fight if you want resources like SP back immediately you can rest, or grab a shroom.