How can I get this elbow off my shower drain, or do I need to replace the entire assembly?
Yes there is a way to do this with out removing the tub. You will need to be very careful doing it though as you could end up doing more work if you do not use care.

Get yourself a 3 inch cutoff wheel and cut a slot on the outside of the elbow perpendicular to the threads, take care not to cut all the way to the threads - you only want to get CLOSE - like a 1/16th or 1/8th away from the threads of the inner pipe. 
You will need to do this in at least 2 places but given that elbow (thanks for posting a picture) - most likely 4 places. Once you do that you can use a pry bar or screw driver to push the lip of the elbow away from the threads - this will break it free enough to allow you to remove the elbow. Kind of like peeling back an onion ...

Remember when you are close to those going through the elbow to the threads - STOP. That way you do not harm the threads on the other pipe. A small nick will not be a problem but cutting those threads might create a leakage issue.