ljw
I'm sorry, but I can't totally answer your question because I am not familar with the MTD built Cub Cadets, or the the 1604.
I can tell you that the 582 Raplh had was an IH built tractor, and had the same transaxle as any other IH built gear drive Cub Cadet. If it would fit in your 1604, then any of them from any model IH built would fit.
If the 1604 is a belt driven hydrostatic drive, putting a gear drive transaxle in it is not going to be easy because of the lack of a clutch and release system, and all the IH transaxles (except for a 482, 582 Special, and 1100) both gear drive and hydro, are shaft driven directly from the engine, with the gear drives having a dryplate clutch between two pressure plates on the engine end of the drive shaft. The Original Cub Cadet is an exception and has a belt from the engine down to a clutch, and then a shaft back to the transaxle. The 482, 582 Special, and 1100 use a shaft from the engine to a right angle gear box, and then a belt with belt tightening clutch to a Peerless type transaxle.
The IH Cub Cadet used the same basic transaxle as the Farmall Cub. Biggest difference is the axles and housings, and the Cub Cadet has a gear reduction on the input side of the transmission rather than in the axles, and the ring gear in the differential is on the opposite side, or flipped 180 degrees.
All in all, it is probably not worth the trouble, but that's just my opinion.