How does comp work at a SWF/PF vs traditional PE firm?

At a traditional PE firm you get salary and bonus as a junior, then eventually some carry as you move up the ranks.

But at a SWF or PF that doesn't raise funds how does that carry part of the comp get reflected for people still in the investing teams?

Now I expect even the likes of GIC/CPPIB/PIF etc won't stack up to MF pay by virtue of just not being direct shops. But would you expect those types of elite SWF/PFs to pay more than a LMM or no-name smaller fund (lets say <$1bn latest raise) if the latter grant carry?

Is it the case that the SWF/PF will pay a much larger bonus to make up for the lack of carry?

Guess the ultimate question is, if I can't make it to a MF, is it more lucrative to go to a no-name PE shop or to a large, well-known SWF/PF, purely in terms of total comp over the years? It's not that that's the only thing I care about, it's more I know where the other factors (prestige, WLB, quality of deals etc) differentiate.