There's no problem running the game on a PS2.
I'm unfamiliar with Duckstore. It appears to be emulation software for the PS3? (Can't find anything about it that's not in Spanish) In the past runners have emulated it on PS3, but It didn't run very well from what I recall.
I don't have a lot to say about splitting the boards aside from I don't believe it is currently necessary due to reason that have been stated earlier in the thread and private discussions.
I am concerned about the loss of the PSP PSN store. (I've checked, it's gone) So yes, currently without jailbreaking a PSP new runners would have to find a PSTV (Not cheap) if they want to run competitively, which I do see as a problem that needs addressing. If we could find an emulator that runs comparatively to the PSP/TV that would be great for a number of reasons, but until then we're kinda stuck. I will note on the Duckstation and accessability, I currently cannot use it because it requires Windows 10 so not everyone can use it still.
I route my PSP into a VCR that I then split to a TV and my capture card. I haven't had any noticeable delay.
I know one of our runners uses (or used) a PSX emulator run on their PS3. We determined it was fine since it basically ran just like a normal emulated run and categorized it as such.