Nope, IMAP can't provide the kind of smooth reliable experience that the Fastmail mobile app provides. Now they obviously could have done it in many different ways, but I know from experience with IMAP that it can't do it.
The amount of times IMAP has resulted in duplicate emails when moving mail between folders for me is enough that I eventually gave up on attempting any kind of organization back when I was last using it.
Anyway, It has been some years since I last used IMAP. I had issues with Outlook, Thunderbird and Nylas N1 while connecting to Gmail, Exchange and Office365. It was a fairly small mailbox getting about 20k mail a year and keeping about half of that. Usually moving about 5k mail around at a time (end of year).
It got so bad that I had to get familiar enough with the IMAP protocol to make my own client program to handle deduplication overnight. Shortly after that I decided to just give up on trying to keep thinks organized.
I am in Australia and this was before the NBN, so obviously it was always on shitty connections.
The amount of times IMAP has resulted in duplicate emails when moving mail between folders for me is enough that I eventually gave up on attempting any kind of organization back when I was last using it.