Wiring question - Swapping in GFCI breaker for my dryer
Go for it

Since you have a full four-wire connection to a NEMA 14-30 receptacle, your plan to replace the breaker with a GFCI breaker is sound, as long as the dryer is properly configured for use with a 4-wire cord.  Note that this will also let you know if your dryer has a ground fault, so you may wish to keep an appliance repairman's phone number on hand in case it turns out your dryer needs fixing.

While you have the outlet box open, it would be a good idea to take a couple lengths of bare 10AWG wire and a wirenut and use them to pigtail the box ground screw and the outlet properly, as well as changing what looks to be an improper ground screw to the correct screw, namely a 10-32 fine pitch machine or self-tapping screw (the latter is available as the Garvin GSST).