Sports transmission? That's new... I'd like more info. I've seen them rebuilt for more aggressive use but never heard of anyone referring to that as a 'swap' of any sort. Sounds more like bullshit to me. Typically hard shifting is due to a problem, start with checking for diagnostic codes and test the TPS (Throttle Position Sensor). See the MKIII FAQ for a thread on how to check for ECU diagnostic codes.
Transmission Diagnostics are here.
I'm assuming this is a N/A... In that case the kick at around 4K is the ACIS system. Inspect the VSV, actuator and vacuum hosing which runs to the center of the intake manifold on the underside.... might as well replace all vacuum tubing while you're at it. The ACIS is a system which uses a butterfly valve inside the upper plenum to modulate airflow and help control torque below 4K. If it's not functioning the car will accelerate poorly (Cressida's didn't have this for example and have poorer throttle response). The 7M-GTE doesn't use this system as it is boosting well before and thus this would largely be wasted on it. I can't find my info on diagnosing the system. I believe that when the engine in in operation racing the engine to around 4K will cause the actuator to activate and you can see that from above, leaning over the driver's side fender and looking down between the plenum and engine (make sure the engine has been run for a couple minutes prior to this, it's not good to race a cold engine).
I'd replace the belt first... It can cause nasty noise and a lot of shops are all too happy to agree with you that something must be horribly wrong. Here's a link to the TSRM's section for power steering:
http://www.cygnusx1.net/Supra/Librar....aspx?S=SR&P=1
That's for a '91 though and there are some subtle differences.... for the most part it'll get you through a replacement of any component. I HIGHLY recommend you buy a copy of Toyota's Service and Repair Manual for your year Supra... it's an excellent resource and MUCH better than others such as Haynes which try to group many years and models into one book 1/3 the size.