Until Frontier remove the ability to choose game mode every time we start a game session, yes, credits and other assets should remain available to the player regardless of game mode.
.... the BGS doesn't care what game mode we are in, nor does PowerPlay (at the moment - it may change, in time, for that specific feature).
If such a change *was* considered, perhaps more important than credits, player rankings with the SuperPowers, Powers, Factions and Engineers would need to be similarly limited - as they can significantly facilitate a CMDR's progress.
Then there's the issue of materials / data - they would also require to be limited to the mode that they were gathered in - as they are at least as valuable as credits.
Which effectively results in what has already been mentioned above: there would need to be completely separate CMDRs for Open and Solo / Private Groups - which would be pointless, in my opinion, as we all affect the same BGS, etc..