Ah ok I think I see what mean about the feeding after short naps. Could you perhaps try resettling (without feeding) for a set amount of time, perhaps 45mins max, and then feed with a small lamp on, but not allow him to fall asleep whilst feeding, and then make sure you take him out of the bedroom for some A time, even if short, before he's allowed to go back to sleep again? Or do you think he's genuinely hungry after his short nap? In that case, you maybe could try topping him up just before his nap, in the room, just don't let him feed to sleep. There's a classic growth spurt at 6mo so it's possible that he may just be more hungry, wdyt? So for example, could you top him up after the morning school run, before his nap?
I'm interested to know how he is after a 20min morning nap? Does he seem really tired afterwards or just happy to play? A 20min nap can be OT or discomfort-related, if the former then hopefully you'd be able to extend it but if the latter he'd probably be harder to settle, or of course if it's hunger due to a growth spurt then a top up before the nap might help. If he's been doing the same A time for several weeks, on the other hand, then he may need it stretched in order to be able to take a longer nap.
I wouldn't worry about feeding to sleep for the last CN, it won't last many more weeks anyway and it's hard to see how he'd get to a decent BT without it. Even once he (hopefully!) starts taking longer naps earlier in the day, you have to fit them between school runs so they may not be able to be quite as long as might be ideal. However having said that it seems that you get longer at home than I do so perhaps that wouldn't be such an issue...sorry, running ahead of myself a bit there
I know what you mean about the school run getting in the way, on the plus side I found that this had the effect of fixing the first nap which helped to prevent EWs from getting out of hand. DD would often end up with a short OT nap but got to the point where she could be resettled eventually, usually with a form of shh pat.
You may need to try a few things to see what works, when it comes to nap extension. Some mums on here have had great success using w2s, you could try going in at say 18mins and lay a hand on him, aiming to catch him at the 20min WU and soothe him back to sleep without him waking up too much. Or you could stay in his room (hide if necessary!) so that you can do this without fear of the door waking him. Or you could go to him as soon as he wakes and do your version of shh pat to try and help him back off again. It's quite common for LOs to be able to ss at the beginning of a nap, but need help to resettle if they do my short nap.
Have you tried an earlier BT? This may just help with the 5:30am WUs if they're due to OT. He certainly seems tired earlier on?