That's good news about your axle.
I can't recommend cutting into the cups due to the "step" in the neck (also keep in mind these races are hardened, and as such, a proper cutting wheel or stone would be needed). Schmidty's idea sounds better imo.
Otherwise, You might also try soaking the cups with penetrating oil (Liquid Wrench, etc) for awhile followed by carefully knocking the cups out with a removal tool. Heating the neck with your lady's hair dryer for a 2-3 minutes can sometimes help. Just work "fast" and careful so that the cups don't have time to soak up the heat (you can try to keep the cups cool with some ice in a plastic bag).
Note: a proper bearing tool is, of course, the best thing to use when it comes to removing and seating bearing cups. Lacking that, try knocking the cups out with a brass punch. Knock them out just a hair at a time starting with the "12 O'clock" position, followed by a just hair at "6 O'clock," followed by a hair at "9 O'clock, and then at "3 O'clock" (repeat this sequence as needed).
Btw, imo the only thing worse than tight fitting bearing cups are cups that fit loose...