If you do the TRI course even as an FO you will get 737 TRI on your licence. If you have not flown 30 sectors in previous 12 months you cannot do the TRI course, only an SFI course. If you do the TRI course Your Licence will be endorsed TRI 737 300-900 sim only.
If you want to instruct on the real aircraft, that requires more training, baulked landings, misshandling etc.
See STD DOC 43 or JAR OPS1
Rank has nothing to do with TRI status