I dabbled with this while in Maine,hunting these critters. The fox we had up there were red fox which are somewhat bigger than a gray fox. BUT......a fox(any fox)ain't got much meat on him for bullet expansion. Even the bigger bobcats are pretty thin.(I killed some that were 45+ pounds in Maine)
I had a 20" Savage 110FP chambered in 223Rem. This was back before the coming of Trail Boss .....however ..... IIRC I ended up shooting a 40gr NosBalTip a bit UNDER 2000 using some "exotic" pistol powder. NO GOOD....the bullet would fail to meet it's minimum expansion threshold and come out the far side. If it struck bone (rib or shoulder)on the way out I could kiss my $30-$100 for the pelt,goodbye. I had better "pelt saving" performance with a 40gr going 3600fps+ out of my 24" AR.
I have no experience with this in the recent years...BUT...I would think that the lightest bullet,driven at the highest speed possible,put square on the shoulder,could work. Maybe one of the powdered metal make-ups if they come in .224".
Better yet -----> Yeah,I know <----- a 17HMR -----OR-----the newish 17 Hornet- --- pruhdlr