bitcoin QT can sometimes use upwards of 1 GB of RAM.
Well there's your problem skooter up there in bold. We are talking about a custom implementation of bitcoind to create a tiny and low-cost bitcoin node. We don't even need a wallet implementation, and we definitely don't need the QT GUI (there's no display lol).
Well most of that memory usage is the db cache. It probably can be done on a custom ARM based board however it isn't going to be a trivial project and you probably want the board to have a gig of memory. The UXTO is only going to get larger over time.