I don't know about the population of feral 'pets', it can makes sense that it's even larger
Anyway both are significantly damaging for different reasons (feral: ecosystem damages and native species reduction, domestic: they are part of the (over)consumerism system, so they virtually increase the human population)
How to cut down both of them? first starting with domestic pets which is the source of the problem, over-consumerism in general
There are going to be a dozen more urgent sources of emissions, environmental damage etc before we start restricting pets. Humans in the developed world are emitting vastly more than their pet -- first discourage the humans over breeding, outlaw religious edicts that make a stand against common sense contraception, etc. Constrain fossil fuel use, consumerism, capitalism.
Thus far it looks like there is little to no chance of any of that happening, so a campaign about pets, carrier bags or drinking straws is rearranging deckchairs on the Titanic.
That's quite a bit removed from pets in countries that control strays, have dog wardens and such. Dogs need no commercial pet food, mainly waste from the huge industrial food globals today -- they will almost certainly do better on traditional feeds -- leftovers, bones, etc, without added oils, colour and sugars.
We are the environment, we're part of it, we're also the solution if we want to be. A better place to start might be to try and figure out how to control a feral dog population of a billion...