"In order to maximize my expected winnings, I should bet 99 times as much on Jenner as Clay."
No, to maximize expected money, you should bet everything on Jenner. Even if if you were 51% certain it's Jenner and 49% certain it's Clay, you'd still need to bet everything on Jenner to maximize expected money. Value of a marginal dollar is proportional to probability of the option, no matter what the current distribution of the bet, so you pump everything in the highest-probability option. If your utility is logarithmic in amount of money, then you bet 99 times as much on Jenner than on Clay (see "proper scoring rules").
"So in this case, a small amount of winnings is less useful than one might naively expect, and the utility function should reflect that by growing faster for small amounts and slower for larger amounts — I should perhaps bet ten percent on Clay. If it’s Jenner, I still get 90% of my stake, but if it’s Clay, I end up with a more-useful 10%"
In order for the bet of 90:10 to be rational, you have to value money even less than logarithm of their amount, which seems implausible. The problem with your original bet was overconfident expression of your state of knowledge (99% certainty), not method of distributing the bet.