Rikki, you can do anything you want to do including using your PSR to play the style and the laptop/synth as your sound source.
And yes you would need a 2nd hard drive if you were going to use very high quality samples as your sound source, e.g., Scarbee 73' Rhodes Piano. The reason is that the samples are being streamed in real time from the hard drive (low latency & no interruptions). This allows you to play a 2.5 GB instrument with only a very small portion of it resident in the computers memory.
The actual playing of the style takes very few computer resources. It is the sound & effect creation that takes all the work.
For tweaking my styles to fit in with the GM standard I use any one or combination of the following:
CASMEdit/Cakewalk
OMB
Midi Player
StyleManger/Updater
I like using CasmEdit and Cakewalk for the midi part tweaking of the style. This lets me get at all aspects of the style (CASM & Midi). In Cakewalk (Instrument Definition File) I can quickly see the instruments that are being used, e.g., Drums, etc. So if the Yamaha Tyros style uses a LiveStandard drum kit in bank 127 and say program number 01 then I can change this to bank 128 and program number 01 (GM Standard). Also with Roland and others you will find that they only respond to drum messages in say channel 10. So if there are drums in channel 9 you would need to assign the appropriate drums or merge it to channel 10. Etc. Etc. It is not that bad once you get a number of styles setup for your system. After it is just a matter of adding new ones as you can use them.
TTS-1 is General Midi2.
And oh Rikki you just can't lose!!!
Just give up on the PSRs, Korgs, Rolands and others. Just buy a dumb controller keyboard and start a new life - exciting - vibrant - creative - ever changing - anything else?