If this scenario really existed, which putt was chosen would likely come down more to personality type and risk aversion more than to how well somebody can putt.
If they don't charge you to putt, there is actually zero risk. You take a putt and if you miss you are no worse off than you were before. But because there is potential money on the table, some people would view a missed putt as 'losing' whatever amount of money was offered. People who view it that way in their heads would be much more likely to choose a shorter putt, because they would see it as less risk of losing the money.
People who took the attitude that it was a no-lose proposition would be more likely to go for higher dollar amounts, because they would be less upset by a miss. Ironically they would probably also be more likely to make any length putt because they wouldn't put as much pressure on themselves.