Wow, sounds/looks great..... all those parts are interchangeable on those bikes. I think from R-75 thru R-100- RS, RT, /7... pretty sure even the cylinders started life the same, just bored out larger... . likely just bolted on the RT fairing...
sir you have a fine bmw r 100 rs with an rt back piece you should be able to check yourself if you can find the serial no somewhere on the frame or engine casing.