luggage – Excess baggage on Vistara (or another airline) + KLM?

Generally you pay excess fees per way for each ticket. So if you have a return ticket A->B->C and C->B->A then you pay excess baggage fees when you check-in for A->B->C. Even if you have to transfer luggage when connecting for the B->C segment, the extra allowance is already paid for. Your bag already be tagged to your final destination or you have to get it at the check-in counter for the B->C segment at which point you should show them the receipt for excess baggage paid, should they not have access to that information.

You have to repeat the process in reverse for you return flight. There could be slightly different agreements in place between airlines which is why contacting them was the right to do. If you tried online contact, you may have better luck calling. Usually though, you are not expected to pay excess fees per segment.