As Harper pointed out, you generally can't status match within an alliance.

LH is also super stingy with status matches, generally only does them under target promotions (IIRC the last one was aimed at MilleMiglia members some time ago).

You can use some of your TAP benefits while earning to M&M (e.g. lounge access), though others will be limited if your *G status is not on your boarding pass (e.g. extra bag, priority security).