On a SRAM AXS MTB drivetrain, can you add satellite shifters on an aerobar?

I am not, among other things, a bike mechanic, and I haven't touched SRAM's stuff. From what I am seeing, this should be possible in theory, but it may involve giving up the Eagle shifter in place of SRAM's clics.

For readers unfamiliar with SRAM, Eagle AXS SRAM’s wireless MTB groupset. I believe that SRAM designed AXS (pronounced "access") a bit like a system architecture. Physical limitations like derailleur capacity aside, components compatible with AXS should work seamlessly together, e.g. some gravel bikes have Force AXS cranks and shifters, but an Eagle AXS rear derailleur and a 10-50 Eagle cassette.

You are correct that aero bars on MTBs are not common, although some long-distance off-road riders may use this setup. Just for general interest, Pete Stetina used aero bars on an MTB to set a Fastest Known Time around the White Rim Trail in Utah, in the US. This is a 100 mile (160km) course that's not friendly to gravel bikes. Of course, he appears to have mechanical Shimano shifting, and hence his setup isn't going to help you.

SRAM offers satellite shifters for its road groups. You plug these directly into your shifters. For time trial bikes without the usual integrated shifter/brake levers, you would set up brake levers on the base bars, place satellite shifters on the bar ends and wherever else you want them, then plug all the shifters into a Blip Box. That item is a bit like a Shimano Di2 A junction box (usually mounted under the stem or in a handlebar end), only the Blip Box communicates wirelessly with the derailleurs. In contrast, the current Shimano Di2 version is wired to the battery and the derailleurs.

This Bike Rumour article about Eagle AXS says that you can replace your Eagle shifter with a Blip Box and a bunch of Blips or Clics - there are several different types of satellite shifter with different model names. The Clics (or at least the ones I linked) appear to be designed for mounting on aero bars. I believe the Eagle MTB shifters themselves lack plugs for satellite Blip shifters, probably because this setup is rare in MTBs. So, to reiterate, I think the proposed setup should be theoretically possible, but I think you would need to ditch the Eagle shifter. That, or find out if your derailleurs will work with both an Eagle shifter plus a blip box wired to your aerobars, i.e. will they correctly interpret the shift signals if both components are installed.