Hi hossit,
I don't know if a SX700 Instrument List is available for Reaper. In principle, such a list is not absolutely necessary for any DAW.
In order for a style recorded in the DAW to be played back correctly via the SX700, it is important that the SysEx at the beginning of the style file is also recorded (and then played back by the DAW). So that channel 9 is also set up for drums in addition to channel 10, so-called "MultiPart Drum Mode SysEx" must be sent in addition to the GM ON and XG ON SysEx. Please see the attached screenshot (= included SysEx in preset style "40sSwingBallad").
You could also first record the style in question with the internal SX700 MIDI recorder. You then import this MIDI file into your DAW. So you have all the style SysEx in the DAW, and the style channels should sound correct again when played back via the SX700 SONG Parts (MIDI Port 1 Receive). And of course it must be ensured that both the DAW and the SX700 have SysEx Transmit and Receive enabled.
P.S.
If necessary/desired, you can of course also control the STYLE parts of the SX700 directly from the DAW (instead of the SONG parts, as described above). To do this, select "Digital Workstation 2" of the USB-MIDI driver as the MIDI output port in the DAW. It is important that you keep the original MIDI channels of the Style recording here (channels 9 - 16).
Hope this helps!
Best regards,
Chris