You can read the floppy in any FDD (mechanically compatible). The problem lies with FDC compatibility.
Each FDC controller looks for a specific pattern on the track to find out where the sector starts are. Some FDCs are incompatible and when using the same floppy formatted on incompatible FDC then no matter what you do you can not read the data.
For example I got many floppies formatted with D40 which is
5.25" drive based on WD2797 FDC for ZX Spectrum. It is incompatible with the PC based FDC NEC765. You can read PC disks on it but not the ZX disks on PC (unless specifically formated on the WD2797 before use).
So you can not read such disks on PC directly no matter what. I am not talking about FAT but low level sync to find sectors FDCs are using. There is an workaround however.
I build custom FDC from AT32UC3A3 which simply is controlled by USB from PC and can seek/read tracks at low level ... extracting pulses. Then send it to PC by USB in form of MFM pulses image. Then it is decoded (both format and FAT and emulator compatible image is created).
I successfully backup all my discs after 15 years of hibernation :) and they where still not demagnetized (even the tapes got unusable a decade before which is weird I always taught they would last longer :) ).
So here are your options:
backup on modern FDD/computer if compatible FDC is present
you should do sector by sector read ignoring the FAT as it might be different then on host OS.
create custom FDC
this is a bit more complex that it sound you basically need to create FDC+FDD+control+data_acquisition system. Mine fits into single MCU chip (no separate FDC or any chip needed just the USB and FDD connectors + some R,C and Xtal. You can use kits for this so you just make a connection cable to FDD and do not need to worry about the MCU circuitry). I would go this way only as last resort. They might be some ready solutions by now like this out there I was doing this a decade ago.
backup on original system
You got BASIC so write program that reads your floppy sector by sector and send the data to PC or whatever by any com available (RS232,parallel port,I/O, sound output, ... ) If none present you can use even the screen+camera create video ant than later analyze it (using as serial "cable")
This is a magnitude simpler then #2. You just need to find the proper com connection that can be used. Read Floppy should not be too hard to code.
So the answer depends on what FDC chip you got in your computer. At HP 9122D service manual you can see the controller PCA part list. Try to identify which IC is the FDC and look for its datasheet for more info about the compatibility. I do not recognize any but I am not familiar with SONY nor Motorola chips numbering and too lazy to search all the candidate chips.
[Edit1] Thanks to @Jules
The FDC controller used should be Fujitsu MB8876A Which is compatible with NEC765 so you should be able to use the simplest method #1.