I think the bidding mechanism is basically sound.
Every player gets a range of tokens in a variety of colours from a limited pool. They offer their to tokens for the right to address the pilgrim. All tokens go into a central pool once the bidding is complete, then they are evened out and passed back.
If everyone has 3 tokens of 3 colours (9 total)
Player A bids 6 tokens (left with 3), Player B bids 4 tokens (left with 5), Player C bids 2 tokens (left with 7).
Once the bidding is complete, 4 tokens are distributed back to each player (A ends up with 7, B with 9, C with 11).
In this way the tokens fluctuate around the group, players might choose to play strategically, offering a low number of tokens for the chance to build up...then launching a massive bid when they see a pilgrim they really want to interact with.
I'm now thinking of a single bidding phase for each pilgrim, because a bid for each question is slowing things down in the test runs I've done so far.
It's getting late, time for sleep. Maybe I'll think fresh in the morning.