Let's study at the museum!
The sentence discussion is currently not working (it's been buggy for the last few weeks and it's now not working at all). So posting this separately.
The given translation is:
Can study and let's do be separated like this:
박물관에서 공부를 합시다.
Or is my construction wrong?
Yes since 공부하다 is one of the majority of 하다 verbs that can be separated, you can absolutely write it like 공부를 합시다. All the verbal effects you would put on that verb goes to 하다 so it doesn't matter, i.e., you could also say 공부를 못 해요. = I can't study.