I realise completely that each barrel is different and you can never tell what will shoot best from each gun BUT

I used to shoot H&N lead free version from a non air arms a while back that worked beautifully. I recently bought a new S410f and tried a couple through that and they hate them.
The reason i need lead free is we have rabbits in the garden, but we have lots of animals so i cannot risk any of them ingesting a lead pellet should it go through and i can't find it etc.
I know there are a few other types of lead free versions, but reading up on them it seems that a lot of them don't like rifles that use a bolt to push through to the breech.
Any advice/recommendations would be appreciated.
Matt