(((hugs))) wilow, I know it's hard when they don't nap well especially when you have an older LO to attend to. Some more thoughts and questions...
There's a classic sleep regression around 4mo, I think some LOs start to experience it a little earlier, some later, so you can expect some shenanigans I'm afraid, but he will relearn in time... You might also need to do some more sleep training, the PU/pd board will help more with this, even if you're veering more towards shh pat than PU/pd. A times in general can start to increase too, although you'll need to be careful of OT after short naps.
Do you have any other prop you can use to guarantee a good nap, say, once a day, if needed? I found the sling to help a lot with this, for example, to keep OT at bay.
It may also take some time to get used to the new cot, is the room nice and dark? If not, maybe he can see more now through the cot bars and find it harder to resettle after a short nap?
When you say he won't resettle, are you trying right up until the next feed, or do you get him up and then feed after a period of activity? If the former, you can consider any calm A time in his cot to count as 'half A time' eg if he's lying there calm(ish) but awake for half an hour, consider it equivalent to roughly 15mins of normal activity. Just a rule of thumb which you may find useful.
After a short nap, he will likely need a reduction in A time otherwise he'll likely do another short nap but OT. How much to reduce the A time is tricky - log what happens each day if you can, and see what works. Up to you how much time you want to spend trying to resettle, but you don't have to wait until the next feed time to feed him, as you say, he'll be too tired by then to feed properly and may fall asleep feeding or just be OT by the time you PD. So don't worry about shortening the interval between feeds. If you're trying to resettle, make sure you get him up with enough time to feed before he needs to sleep again. If you get him up straight away, you might find you can put him down again for another CN in say, 20-30mins and feed afterwards, although I personally haven't had experience with that approach. I guess I just want to say you can be flexible with feed times, perhaps think of feeding every 2.5-3.5h (for,example), rather than strictly every 3.25h. Equally if he ends up doing a long catch-up nap then you can let him nap longer, I'd usually cap at 2h at this stage, although I have let mine do 15mins more at times.
On play school days can you put him down a bit earlier for his second nap? I see your routine for non-playschool days has him asleep at 11.20 for an hour, but you put him down at 11.45 on playschool days? Having said that, has he been having a 2h nap on these days in the afternoon most recently? If so, that's brilliant!
I see he's been having more trouble at BT, I suspect this is OT from all the short napping and resulting long A times - reducing A times after a short nap should help with this and again, don't worry about set BT at this stage, he may end up going to bed earlier if he's been short napping and this is fine as he'll need to catch up with lost daytime sleep during the night.
Does playschool run throughout the summer or do you get a break? If so, how many weeks left before the break? Just wondering how temporary this situation is - if you break for the summer then maybe he'll be ready for a nice long morning nap at home by the time your older LO goes back, which is lovely once it happens
Sorry if this is rambling, had dinner in the middle! Let us know your thoughts and we'll walk through this with you
