Even if you could get over the logistics of shipping in a mini TBM, building it and establishing power, water supply etc at the same time creating a break in point for the machine, usually a shaft to the required depth of the machine with a balk wall they can jack off for launch, you'd be looking at around a month, probably two, before they could get going and then you'd only progress at 100-200m per week so you'd be way beyond the period for just waiting for the rainy season to finish and the caves to dry out normally. On top of that depending on the strata of the rock/soil an EPBM machine might be the only one capable of drilling and that could potentially collapse or flood caverns with the pressure balance if they hit fractured rock or cavities.
Best mechanical bet would be to drill vertically from above as they did with the Chilean miners but again that really depends on how deep their location is from the surface, anything more than 100m is pushing the bounds of the biggest drilling rigs and really looking more like vertical tunneling with all the previous issues.
If they can't swim/tow them out then sitting tight may be the only safe option as horrible as that souds.
Thanks for the engineering perspective BR. Have a feeling they may have to wait for it to dry up a bit. Sounds too horrendous for young inexperienced kids to attempt, hope I’m wrong. Wonder where they are going to toilet