I personally really appreciate having some physical amps around for my own pleasure and reference against my modelers. If you are comfortable with with only 1-2 sounds, no loop effects, etc, then the 2203 makes the most sense to me. It's going to simply do the thing you love. If that's the sound you need, go straight to the source. The Axe FX really comes in handy when you need more variety of tones, or more complicated effects routings, etc. Honestly, the Axe FX with poweramp, controller, etc. is at minimum as cumbersome as a traditional rig, and in some cases, even more so.
I vote amp, but consider your needs in a band/show setting. If you buy the amp, are you also going to have to buy an attenuator, additional pedals, more cables, etc? Factor all those costs in when comparing against the Axe FX III, which you would also need a poweramp with unless you do the FRFR thing. If you're 100% content with plugging an overdrive straight into a 2203, then it seems silly to spend more on a Axe FX rig to replicate that same setup digitally.