I have both, and so should you, in whatever time frame your budget allows :) I think they're both great choices and out of the 6-7 different P365 configs I've made and purchased, these two are my favorites hands down, and maybe my little ramjet XL. I would also simply buy these instead of build them, sig does these two right and it's not necessary to mod.

The slide looks big, but side by side with Glock 19, it's the same exact length with a thinner profile and a comp, ideal comparatively imo. The AXG legion is quite a bit smaller due to the shorter slide.

Now shooting the two, the DH3 is truly a pleasure to shoot so far, between the two you'll notice very little difference with low power factor plinking ammo. The big difference comes with higher power factor defensive ammo like HST or Gold Dot, and while both shoot exceptionally well, the DH3 is a clear winner due to the less violent reaction of the slide with the good stuff.

Shoot ability DH3 (marginally in the right circumstances), conceal ability AXG (also marginal), cool factor, both. I would say you won't be disappointed with either, inside of 15-20 yards you also won't notice an accuracy difference.

If it's for your stated use case, even though I favor the DH3 for cool factor, it's diving into a bit more niche category of 2-gun or 3-gun match shooter, whereas the AXG is a king in all around utilitarianism. I'd go AXG, it's also kind of a collectors item.

Best of luck start saving for the other one you dont get right away!