Is there anything in the NEC that would prevent me from attaching MC cable with ground to EMT with a compression coupling?
The only thing I could think of that might be wrong with that is the wires in the MC may not be listed / marked as THHN or THWN, even though that's what it is.  

Could you use greenfield (FMC) and pull in your wire?  

Regarding the coupling - I think you'll have to use a MC/AC box connector and EMT box connector joined together by a female-female threaded coupler.  

edit:  Actually it looks like there are EMT to FMC couplers:  

http://www.bptfittings.com/Home/ProductDetail?s=582-DC(Catalog%20Num)