I feel your pain. Part of the reason I switched from the Proxxon to the Sherline lathe was exactly because I could not get a self-centering four-jaw chuck for the Proxxon. I went down every rabbit hole suggested by contributors at MSW, and looked at countless YouTube videos explaining how you could get spindle adapters to fit standard available four-jaw chucks to the Proxxon proprietary spindle, but ultimately gave up.
I'm given to understand that the micro mark lathe use standard sized fittings. You should be able to find a 4-jaw chuck and, if need be, a spindle adapter for it.
I asked Gemini, and the A/I agent agrees.
"To mount a 4-jaw self-centering chuck onto your Micro-Mark MicroLux 7x16 mini lathe (Model 82995), you will need a compatible 4-inch plain-back chuck and a spindle adapter plate (backing plate) designed specifically for 7-series mini lathes."
SoS deleted the link to Little Machine Shop for the recommended chuck. Confirm the diameter and number of threads on your spindle, and shop four jaw chucks at Amazon.
I asked Grok the same question, got a similar answer. No adapter should be necessary because a 1 inch by 8TPI spindle nose seems to be the standard. I therefore reiterate, check Amazon for self-centering four jaw chuck's, check the specs to make sure they have one inch by 8 TPI spindle.