Load testing seems attractive, because after you use one of the remedial approaches, someone is likely to say, "yes, but how do you know you gained the required additional capacity?".
You haven't told us what kind of a building it is, but perhaps you can load test with dead load such as pallets of material or water tanks clustered around a column. If not, perhaps you can test with tension piles (screw piles, maybe?) and a pair of beams clamped to the column or running across the pile cap. You might want to shore the first-floor beams in case the foundation wants to plunge, but that seems unlikely. How much load would you need? Would you have to prove every pile group, or just one or two representative ones?