Yes, fix it to the studs, NOT the lath

Sand

Lath and plaster construction from the 1930s might contain asbestos, or it might not. It could have animal hair mixed into the plaster, instead. The only way to know whether a particular installation contains asbestos is to have a sample collected by a qualified technician and analysed by a qualified laboratory.

lath and mortar walls and ceilings have a greater flexibility than standard plaster. If you do plaster over it you firstly need to be very confident that the old mortar is sound and fully adhered to the lath. You would also have to seal the ceiling well to stop the water in the plaster from literally being sucked out of the plaster. Regarding tiles. If you are thing about ceramic tile then NO!. The weight of the tiles would pull the mortar from the lath with great ease. It is defiantly better to either plasterboard directly to the old ceiling and using dry wall screws NOT NAILS screw directly to the joist not the late you can then put a good scrim tale over the joints and plaster accordingly. The very best option however, is ...............drop the old ceiling, re-board and plaster

Squirt on vinegar and water (1:1 mix) with a squirtbottle and leave overnight.