I've got all the 3 brands that you mentioned, mine are sd2, pa800, psr1500.

First up, sd2 is really designed to work with a pc. Doesn't have any buttons or screen, so you need a controller that will send the correct bank & program changes.

Tried it with my psr 1500 initially, it made for a lousy controller keyboard.
I couldn't put in the bank changes for melody parts, so, I was stuck with using the default GM bank.
I was able to use the sd2 as a sound module for psr style parts, but there was no way of choosing which tracks were internal ( psr sounds) & which tracks were played by the sd2.
Basically every style I wanted to use became a user style, because of the editing required. XG mapped drum tracks aren't good when played on a GM module, they'll need some remapping..

My Korg PA800 on the other hand is a good controller.
I was able to put in the bank & program changes for melody. I could individually pick which tracks used internal or external sounds.
But if I literally wanted the sd2 to be the sole soundsource for the korg styles, again the styles would need to be edited.

Before selling my sd1+, I tried using the sd2 as the soundsource for the sd1 styles.
Even they would have required some tweaking.

Try & get a keyboard that you like the sounds as well as the styles on. Trying to use an alternate soundsource for "style parts", means there's usually a certain amount of tweaking involved.

I used 3rd party software to edit the styles on my psr for use with the sd2.
Can't remember to what extent you can edit onboard styles in a psr, haven't used it in ages.
On my PA800 all style editing can be done on the keyboard itself, though I do use a pc for certain things.

I don't bother midi-ing the sd2 to the korg anymore, more than happy with the internal sounds.
Only thing I may consider down the track is a vsti module.

best wishes

best wishes
best wishes
Rikki 🧸

SX900, Kawai es920 piano Band in a Box 2021
V Arranger/Ketron SD2