"The Mac hasn't been tampered with but it thinks there is no internal hard drive. " So I assume you are the one who moved the OS.
It's your Mac so I'll go with that, but again, something is not right in the usual sense.
My next thought is the internal drive has suffered a catastrophic failure. If you can't see it in Disk Utility, can't boot from it and can't select it as the startup drive I'd assume it's damaged beyond recognition much less repair.
If you wish to attempt to salvage the disk, I suggest you get it to Apple for an evaluation. But depending on the age of the Mac, you might be in a situation whereby the cost to repair, if they agree to repair it, may be greater than the value of the Mac. You've yet to say how the Mac is.
If the internal disk is dead, you've already solved that problem with an external boot drive. Leave it alone and continue to start up from it. That is a very acceptable solution.
"Unfortunately, Adobe Lightroom doesn't work with this new setup, and I've tried everything" This should not be happening. Adobe doesn't care what disk it runs from as long as there's sufficient disk space, enough RAM and it's properly licensed if that's required.
Since you appear to be limited to the external boot drive, I suggest go with it still and contact Adobe for tech support. They do not need to know what drive the app is running from to help. Well, we haven't used Adobe in a very long time so for all I know they may have made a genius move to that effect. But I doubt that, they want the business too badly.
All this is just my solution and comments. Believe me this community is filled with members far more knowledgable than I am, so I'll just stop here, I've said all I can and there's no need to repeat myself, and let others take a crack at helping you.