Lots of questions/possibilitties come to mind...
It would help to have an exact model number so we could look up the specs...
Does the HD "spin up" - often the prolem with such old HDs is "sticktion" which you can somtimes get working by rotating the drive sideways (along the axis of the spindle) fairly aggressively during power-up.
I also had a drive once that I "took apart" and manually freed the heads from the platter in the hopes of it working long enough to recover data from it - which then continued to work reliably for years. Always best to do such stuff in a clean-room, but just keep the work area spotless, I don't think you have much to lose.
Really worth trying to get a working HD - trying to run from floppy is quite limiting - and you end up putting a lot of wear and tear on rare/difficult to replace items.
Failing that if there's enough RAM you might be able to make a boot floppy that essentially runs the system on a RAMdrive.
ATs(286s) often had 5.25" HD (1.2m) drives ... are you sure the drive in it is 360k (the AT could access/boot 360k disks on an HD drive) - if you can get ImageDisk to run on it, you could tell by stepping 40 tracks from home and seeing if the head ends up near the other extreme (360k) or near the middle (1.2m)
Can you enter a BIOS or have a "setup disk"? If yes, what does it show the drive as configured as, and does it allow 5.25" HD or 3.5" HD to be configured?
If yes to either, it has an HD floppy controller, and you could probably replace the drive with a GoTek - GoTeks are often used to replace 3.5" HD drives. but I think I tested mine and confirmed that it could be used as a 5.25" HD. (A GoTek would also solve the wear problem, and there are tools to access the USB stick you use in one from another system which would make transferring stuff possible).
You say you would normally make 3.5" disks - sounds like you have another system with a floppy/controller - you could pull the drive from the portable and use it to make disks by writing 360k images, or just access existing disks to put files on/off of them. All you might need is a floppy cable with a 5.25" connector (the one in the portable might work).
Does the machine have a network port? If yes, we might be able to find a "packet driver" to access it and you could use my DDLINK to moves files to/from it. If not network, does it have a COM port? - You could do the same using that interface (just slower).
Just some ideas to try and help make the machine more usable!
- Dave ; https://dunfield.themindfactory.com ; "Daves Old Computers" ; SW dev addict best known:
ImageDisk: rd/wr ANY floppy PChardware can ; Micro-C: compiler for DOS+ManySmallCPU ; DDLINK: simple/small FileTrans(w/o netSW)via Lan/Lpt/Serial