How about final salvo bids? Everybody gets (rules) dice, and get an extra for every point under 200. That gives you ~10% better chance of moving last for each point, but weird diminishing returns for digging too deep, and those first five points or so are very important.
I like that use of salvo better than tiebreaker.
Edit: new page context “Is there some way to combine bids and randomness? Bid--with diminishing returns--slightly improving a dice roll-off for example.”