Your definition of total pre-order:

A total preorder % satisfies the following properties: For all x, y, and z, if x % y and y % z then x % z (transitivity). For all x and y, x % y or y % x (totality). (I substituted "%" for their symbol, since markdown doesn't translate their symbol.) Let "A %B" represent "I am indifferent between, or prefer, A to B".

Looks to me like it's equivalent to what I wrote for rule 1. In particular, you say:

To wit, I am indifferent between A and B, and between B and C, but I prefer A to C. This satisfies the total preorder, but violates Rule 1.

No, this violates total pre-order, as you've written it.

Since you are indifferent between A and B, and between B and C: A%B, B%A, B%C, C%B. By transitivity, A%C and C%A. Therefore, you are indifferent between A and C.

The "other" type of indifference, you have neither A%B nor B%A (I called this incomparability). But it violates totality.

I don't think Rule 1 is a requirement of rationality

Hope you'll forgive me if I set this aside. I want to grant absolutely every hypothesis to the Bayesian, except the specific thing I intend to challenge.