Flakes of putty is clogging the bathtub drain. Is there any product to dissolve the putty?

Are you sure it is the putty that is entering the drain? There is very little of putty under the tub drain flange. Even if it got inside the drain, it would be flushed down the drain with draining water. But most importantly, the putty used to seal drain flange is on the outside of the flange, not on the inside. So not sure how it gets inside the drain. Please, re-inspect the drain. Maybe the debris is coming from some other source. Can you snap a picture of it and post it for us ?

Hope that helps

Milo