Well I tried using dd to write the Lemmings image back to another disk and then using that disk in a real machine and Lemmings could tell it wasn't the original, so it's probably a limitation of the .img format.
I also tried mounting the real floppy in DOSBox, but as wd mentioned it isn't recognized as the original because DOSBox is unable to read the real disk at a low enough level.
It's all good, though, because I can crack it.
Edit: Well I think I've made it through my 3.5" floppy collection, which is much, much smaller than my 5.25" collection (both in terms of original game disks and miscellaneous personal disks). This is because I was without a 3.5" drive for a very long time, and once I did get one it wasn't long before I also got a CD-ROM drive. I've attached my 3.5" ripping notes/list in case anyone is curious.
A few game disks were sadly lost to the ravages of time, with issues ranging from a bad sector in a file to being completely unreadable. The last disk of The Two Towers had a bad sector in the empty space right at the end, so I just copied off and zipped up its file(s) instead of making an image. Fortunately, I think I have 5.25" disks for most of the games with bad 3.5" disks, but I don't know how well 5.25" disks generally hold up compared to 3.5" ones (hopefully better due to the lower density).
I also skimmed some miscellaneous stuff off of non-game 3.5" disks, including a half-dozen TSRs that let you run CGA programs on Hercules video cards, a backup of First Publisher, some MIDI and MOD music, and a whole bunch of backed-up game save files.
Total 3.5" haul was a little under 30MB. I expect to get several times that from 5.25" disks though. My goal is to put it all onto one CD-ROM as well as leaving the files and images on my 2TB hard drive (that I use for storing videos, ISOs and other backup/archival stuff) so that I can access them from DOSBox.
Edit 2: Oops, found more 3.5" disks in the garage. Will update my list after I go through them.
Edit 3: Finished ripping this second cache of 3.5" disks and updated the attached list. Will probably tackle the much larger 5.25" collection over xmas break in a few weeks.