In a past life we used OR-Tools for a problem of assigning data shards to serving tasks, where the data shards had heterogenous demands (e.g. some shards were low traffic but demanded sub millisecond latency targets and thus were served from RAM, others were higher traffic but could tolerate being served from flash, etc.). It's insane how expressive this thing is! But the problem got to be so large that we ended up having to hand-roll something less optimal because it would take multiple minutes to generate assignments -- think: millions of shards, tens of thousands of serving tasks, and I want to say it was ultimately nine dimensions of constraints.
You may have tried this already but often times systems require things to be sticky (ex: to increase caching efficiency) and that usually helps solving large problems since most solvers accept "hints" or "warm starts". CP-SAT does a great job accepting hints and cuts down the search time significantly if the hint is good.
"Multiple minutes" doesn't sound like a lot. With millions of shards, do you really need to regenerate the assignment layout every couple of minutes?