Keep in mind that the FM3, FM9, and Axe III will not sound identical due to physical (components) and software differences. They're about 98% the same, assuming all of the available parameters on the FM units are set the same as the ones on the Axe III, but again, there are some settings on the Axe III that simply don't exist on the FM units that can affect how it feels and sounds. Somehwere there's a list of these differences. It may or may not make any difference to you, but it does to me. I cannot take a preset from my Axe III and use it on my FM3 and expect it to feel and sound the same - again, it's nearly identical, but not entirely.
My advice would be to get two of the same unit, whichever you choose. If you can abide the nuanced differences, then I'd get the flagship for home and the FM9 for gigs if your presets or switching needs are more extensive, or the FM3 if they're not. Really, you can't go wrong any way you go.