Here's a thought.
To whom is this petition going to be handed ? If some EU body then they already have legislated against the practice.
"Within the borders of the EU, birds have had special conservation status since the creation of the landmark Birds Directive in 1979, a policy designed to protect the populations of all bird species from hunting and habitat loss"
So if that's not acted on what will ?
If it's to the company then they are unlikely to stop selling if it's lucrative. (And threatening to stop buying their cartridges if they don't, is unlikely to affect them much.
If it's the UK government, well how much songbird shooting goes on here ?
Outside of those option no one will take a blind bit of notice. So I'm wondering what the plan was for it.