Has anyone managed to get an LMT eSIM working on the MikroTik SXT LTE6 kit?

You can buy 9esim adapter.

Will the eSIM not work without this adapter?

No. You need some "hardware" to enable eSIM. The older devices do not have the need things on the "motherboard" or modem module, so only choice is to use a special physical/hard SIM card that will allow RouterOS eSIM commands to work.

If you're not planing on switching carriers, you can just a physical SIM from LMT - this avoid needing any special "physical eSIM" like the 9esim SIM card – but you're going to need SOME SIM card installed for the modem to work.

There is longer discussion about this topic here:
viewtopic.php?t=217140