Sending the familiar to its pocket dimension doesn't help.
Making the familiar go to its pocket is merely movement and has nothing to do with the familiar's physical condition. It doesn't regain HP or shed any conditions when this happens. It might be more convenient to carry around a petrified animal by tossing it into an extradimensional space, but it won't solve the root problem.
Casting find familiar should fix the issue.
The find familiar spell says:
When the familiar drops to 0 hit points, it disappears, leaving behind no physical form. It reappears after you cast this spell again.
So if it's killed, the familiar leaves behind no physical form when it vanishes, and when it reappears, the clear intent is that it comes back with a brand new body at full HP, bearing none of the injuries that killed it.
If it "reappears" without any injuries, I think it's logical to assume that any other effects attached to its old body are similarly left behind, whether that's poison, a curse, or petrification. (Or, for that matter, blessings and other beneficial effects.)