I don't own L3m or any StageSource speaker, but I might have one idea. What Smart Speaker mode are you using when you have HD500 connected via L6Link? Maybe if you use Electric guitar mode, it tells HD500 to switch to Combo Pwr Amp for some reason and if you change mode to Reference/P.A. it might change to Studio/Direct, but again, I don't have L3m, so just one quick idea...
Is your HD500 "wearing" the latest firmware? Stagesource compatibility has just been added to the HD with the last version, Maybe the problem is that your HD500 is seeing L3m as a DT amp because it doesn't "know" about Stagesource.
The volume changes in live mode are normal. I wish it wasn't. I wish the amps, cabs and mics were "normalized" in volume, so that switching between them would only change the tone, not the volume, but they aren't, so, if you want to use, say, the same patches when you go with direct/studio mode as when you use any of the other modes, you'll have to do different setlists and recalibrate volumes.
I found about this the hard way, that is, huge volume raise when changin patch to play a solo. Pretty embarrassing