To get to Bayes, don't you also need to believe not just that probability theory is internally consistent (your well-ordered ordinal gives you that much) but also that it is the correct system for deducing credences from other credences? That is, you need to believe Cox's assumptions, or equivalently (I think) Jayes' desiderata (consistent, non-ideological, quantitative). Without these, you can do all the probability theory you want but you'll never be able to point at the number at the end of a calculation and say "that is now my credence for the sun rising tomorrow".