I talked to TravelPerk again, and this is basically what they said:

The platform mainly features third-party rates that don't count toward loyalty status. But they insist that they DO have bookings that respect loyalty status (probably to dissuade me from not using their portal).

However, apparently TravelPerk doesn't show which rate it is booking under the hood. Instead, the way to find eligible hotels is to apply a less-than-obvious "Loyalty Chains filter" after executing a search. If you don't see your loyalty program there, then they don't have a rate in which it can pass on the loyalty status. Kind of an annoying UX in my opinion, but at least now I know.

So basically (my takeaway), always assume the loyalty program doesn't get passed on unless there's a "Marriott Bonvoy" filter active (and in some listings it will say "Earn Marriott Bonvoy points!" on the booking).

I hope this information helps others struggling with TravelPerk.