When my 286-XT arrived it booted fine too, but it was really just sucking the very last of the magentism off the disk, 3 boots later and it couldn't see anything - did a low level format and viola as good as new.
For getting data on to a machine, null-modem cable for the win. In fact I used it yesterday to load up my desktop 386 with goodies (needed WFW311 to get on the network
). I got my cable delivered next-day for around $15NZD (10US?), so they're still out there, and easily made if you're so inclined.
I use FastLynx 2 (FX2). It has an upload mode which can copy itself to a host thanks to the DOS command 'ctty'. So you don't even need to copy it on to a disk first. It's an easy to use uttility with built in help etc. It usually works even if your DOS folder is missing, because it uses the same defaults for uploading as a standard DOS boot (2400,n,8,1)
Usually "ctty com1:" or "ctty com2:" is all you'd need on the client, then select Upload from FX2's menu on the host. (and I beleive ctty is part of command.com).
Once it finishes I find I have to reboot, then load it up, and you can use either machine to transfer/delete files.
FX2 and a lot of the DOS utils for drive formatting etc easily fit on floppies.
Edit: the above assumes you have at least one other machine with a serial port. If not, you may need to get yourself a hack machine to bridge the gap. Parallel is an option but can't remember if ctty works with it, FX2 does.
Edit2: if you fit that AMI BIOS you mentioned, I'm fairly certain you can Low Level Format with it
Edit3: I'm also assuming you have a spare disk with space on it - FX or FX2 are quite small but not tiny, in the 150Kb+ range from memory.