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I haven’t mixed and matched on my road bike, but I use a shimano, chain, cassette and chainring with an axs mech on my mtb and it’s flawless. Arguably better than what it was with a full sram set up.
I imagine axs road mechs with 12 speed shimano cassette, chain and chainrings would be fine.
FD shifting might not be ideal though due to the different tooth jumps from big ring to small ring.
really interested to hear how you get on. would like to go AXS but the cost of chain, crankset, cassette really really put me off....!