I think OpossumJoker would hide absolutely nothing or extremely important bits of stuff in his pouch so Batdog would need to dig around in there.
I think he would do it because 1, it's amusing to see Batdog be so cautious and shy, and 2, he likes feeling Batdog's paw in his pouch.
For Opossum Joker it's just another part of his body, nothing worthwhile to think of in any other matter than 'place of storage' for funsies whereas Brucedog takes it rather serious. He's more concerned on the aspect that Joker is a male that was born with a pouch, obviously a medical interest (aka science experiment #1) to anyone in Arkham who may take advantage of their doctor status to do more than they should.
Opossum being an orphan probably opened him up to a whole lot of other children who would ask him about it or randomly poke. I wonder if Batdog didn't touch the pouch at first but then when he dropped him off at Arkham and the screening process caused Opossum such distress he went into full aggro mode which made Batdog go 'aight I'll be checking that next time'.
(based on @koshikakuri's comic)