With the amount of cancerous plastic and preservatives in all the other food and water you consume, lead in game meat is an absolute nothing burger.
besides, if you ate so much game meat you got cancer from the trace amounts of lead in it, you lived a great life anyway.
Suppressors are awesome. You don't need to justify buying one to me at all. With that said, I've killed a lot of multiples without them. They are hearing your shot regardless, although it may spook them more without, who knows. If you have the extra coin and time to spend on the suppressor...
I have an older VSSF in .220 swift that is really my main rifle. It is sub Moa with nearly every handload I feed it. I think a .223 in one would serve you well.
the 52 grainers work well too. they werent as consistently fur friendly as the 53 grainers though. If you dont reload you may want to try these.
https://www.brownells.com/ammunition/rifle-ammunition/223-remington-52gr-match-hollow-point-ammo/
Do not use FMJ's. There isn't any consistency in their terminal performance in my experience, If you hit bone they create a giant mess and if you don't they zip right through without creating the damage you need.
The best bullet I have found has been the 53 grain Sierra Matchking. Foxes are...