

I’m not much of an expert on Bluetooth, but I would expect that you can create an override for the corresponding Systemd service (bluetoothd
perhaps, or some Logitech daemon) and make it depend on a Target that is reached earlier in the boot process.
Sorry that I can’t be more helpful…
I run a J5040 ITX board for my homelab needs, which has been released a few years ago and has served me well, even through I run it with more RAM than the board specs allow. The natural successors of that are the Atom N100/N105 and the i3 N300/N305 (all 1 Gen newer than J5040) and AFAIK the Atom N150 and i3 N350 (2 Gen newer), all of which are available on ITX boards. Models for the latest chips might be a but rare though, and you might have to go to AliExpress to get one, but for the N100/105/300/305, there’s a wide variety available. Just make sure to get one with enough SATA ports for all your disks, so you can use it for NAS as well.
Disclaimer: I’m quite sure this is enough for your homelab/NAS use-case, but I’m not familiar with Minecraft requirements, and you might need beefier hardware for that. However, the above boards leave enough room in your budget for RAM, NVMe and HDDs, should deliver quite some bang for the little buck you have, and will barely sip energy, making cooling easy.