In the same way that we work with delta functions as though they were ordinary functions, we can work with lexicographic preferences as though they were ordinary utility-function preferences.

There's a pattern which comes up in math sometimes where some rules hold for all the functions in some set, and the rules also apply "in the limit" to things "on the boundary" of the set, even if the "boundary" is kinda weird - e.g. a boundary at infinity. Delta functions are a prototypical example: we usually define them as a limit of ordinary functions, and then work with them as though they were ordinary functions. (Though this is not necessarily the best way to think of delta functions - the operator formalism is cleaner is some ways.)

Lexicographic preferences fit this pattern: they are a limiting case of ordinary utility functions. Specifically, given the lexicographic utility sequence $(u_1,u_2,...)$ and some real number a, we can construct a single utility function $u_a:={\sum }_i{a}^{-i}{u}_i$. In the limit as $a\to \infty$, this converges to the same preferences as the lexicographic utility. So, just like we can work with delta functions like ordinary functions and then take the limit later if we have to (or, more often, just leave the limit implicit everywhere), we can work with lexicographic preferences like ordinary utilities and then take the limit later if we have to (or, more often, just leave the limit implicit everywhere).

To answer your exact question: lexicographic utilities are consistent with the coherence theorems, as long as we drop smoothness/boundedness assumptions. They shouldn't be excluded by coherence considerations.

If one uses surreals for chances then one can provide the required chance to make continuity happen.

There might be multiple ways to map a lexiographic weight to 1 + w + w^2 + w^3... but I would expect that simiarly that real functions can be scaled for no effect using different transfinites would just be a matter of consistency. Ie whether you map (1,2,0) to 1+2w or 1+ 2w^2 is a choice that can be made freely if it is just sticked to.

Then you can have a 1/w=e where 0<e<1 which can function as the p to make the lottery mix exactly ambipreferable.

I imagine you would be able to solve this by replacing real-valued utilities with utilities in a numbering system that contains infinitesimals. However, it seems to me that it would not matter much in practice, since if you had even a tiny chance of affecting the leading term, then that chance would outweigh all of the other terms in the calculation, and so in practice only the leading term would matter for the decisions.

Or to give an argument from reflection, there's a cost to making decisions, so an agent with a lexicographic ordering would probably want to self-modify into an agent that only considers the leading term in the preference ordering, since then it doesn't have to spend resources on the other terms.