There's a few things you could do.
Install an extra virtual floppy drive and mount the image there. VFD is very handy for that. Available for free at http://chitchat.at.infoseek.co.jp/vmware/vfd.html
Does the game check if the image is formatted? Or does it just format the image no matter if it's already formatted and empty? Considering your issue I suspect the latter.
This problem could also be solved by hooking the disk read/write BIOS functions the game uses. If you have a formatted and empty 360K image ready then it should skip the formatting process anyway. But it may do some game unique stuff to the image that makes it valid for the game only. Normally hooking the functions and returning successfull completion of the function should do.
I'm also wondering if the error might be caused by you swapping the disks. Afaik DOSBox doesn't refresh the mounts directly. It could also be that DOSBox hasn't read in the new mounted content yet. How exactly do you perform the mounting? With the VFD driver you could mount it before you start the game and swapping wouldn't be required, unless the game forces you to use the same drive.