How much health do sleeping monsters regain?
Solution 1:
That's exactly it - they regain enough to get out of the limping/capturable threshold. This generally corresponds to around 25% of their maximum HP for the later monsters (anything past a Royal Ludroth) and 40% or so for Ludroths and bird wyverns. Note that attacks on a sleeping monster gain bonus damage, so hitting them while sleeping will push them back below the capturable threshold.