Short version: I don't know. But I can say that space-based lasers have a history of flight anomalies. ICESat-1 being a prime example.
In that case, the failure review board concluded (link below):
...that the most likely cause of failure of GLAS Laser 1 was an unexpected failure mechanism in a pump diode array that resulted in excessive power degradation ...
Yes, if the cavern and the mineshaft above it could be filled with perfluorobutane, the cavern could be habitable because that inert greenhouse gas is 5.4 times as dense as CO2 !
To add some sense of reality I've situated the shaft in Nier crater (43.1⁰ N, 106.0⁰ E) because, according to the NASA's Treasure Map for Water Ice on Mars, it has subsurface ice ...
Looks like it's not possible:
To get Earth-normal pressure we need 55km
But the deepest we can go looks like 7km
What's the deepest a trench could theoretically be dug on Mars?
And even if lava isn't a problem the rocks ...