One possibility might be to just take carry-on yourself and ship the rest of your luggage separately to your final destination. This can be expensive, but if you really save a lot of money by buying tickets on separate airlines, you might still come out ahead.
Has anyone actually tried this? Can you pick up the shipped luggage at your final destination airport and take it through customs, and will it be reliably there? Or will this all be too uncertain or too much of a hassle?