Have you yet found an example where the (apparent) optimal solution requires dividing any stick into more than three pieces?

No, but I haven't looked at any especially large examples, and that's where I'd most expect to see large dissections. I was rather hoping to find a sensible search algorithm and then run it on lots of pairs to collect data for a conjecture, but so far I haven't thought of a good criterion for the search algorithm figuring out when it can be sure all subsequent solutions it finds will be worse.

Is there any good reason not to consider wlog m<n?

None at all – the problem is certainly symmetric in m,n.