Is anything assigned to drive B in win98?

Google AI (a google search) actually provides a three method answer for this.. "win98 manually assign drive letter floppy drive"
Method 1. Device Manager-> Floppy drive -> Setting tab -> letter assignment.
Method 2. Physical cable: Drive A = after twist. Drive B = before twist.
Method 3. Third party tools."Letter Assigner" by Vadim Burtyansky

External drive bays makes me think of usb or a add in card which quite likely goes to E.

Old documentation says IRQ6 might need to be released

In your bios is the floppy controller or drive disabled or conflicting with something else?

I agree.. "external bay" sounds suspect, but BIOS "swap floppy" working makes me thing it's a standard connection to a mainboard floppy controller.
Can you tell us exactly what the drive is connected to and how? Is it a mainboard floppy drive connector, via a std 34-pin floppy cable.

Obviously something is tying up the B: position, the trick is to figure out exactly what and how.

Some things to try:

Disable drive B: in BIOS - does it "disappear"?

If you go to DOS prompt and type "DIR B:" what exactly does it say?

Restart in DOS mode, and try the same.

Boot from a DOS floppy (or image on GoTek) (easy to make if you don't have one), and then see if drives are A: and B:
(just reread your post and see that you've done this - see what "restart in DOS mode" does)
The fact that DOS alone see A: and B: correctly suggests that the connections are right and no hardware conflicts - my guess is that Winblows is taking something over...

If not, remove HDD and repeat.

sigh.

@DaveDDS @rmay635703

what is meant by "external bays" in a computer case is that they are accessible from the outside without opening up the case in contrast to internal bays for which you would have to open the case.

What happens when you boot to a command prompt (from the F8 Windows boot menu) or into safe mode?

Did you install any of those unofficial Win98 update packs?