Rusty,
from south of the equator I'm not sure of the exact timing of the breeding season of your tits so my suggestion might have to wait until any nestlings have flown. A bit of smoke blown into the hole will cause most birds/rodents/etc to vacate the premises rapidly, if they can do so. A bit of burning rubber smoke increases the effect while still staying humane. Deserted eggs won't smell much but you might still want to wait until after the season. The smoke can be blown along a length of tube so that it doesn't represent a barrier between where they are and the exit.

Cheers, Rowan