We have 4 cats, 3 of whom are hunters. Occasionally we get a bird, however mice, shrews and slow worms are their preferred prey.
Ultimately, cats hunt whatever is in greatest abundance and the only answer will be to relocate your feeders to an area that is not sheltered by plants, so the suggested middle of the lawn could work. It's going to be a case of trial and error.
We don't feed the birds for this exact reason, and we're sorry that it wouldn't be kind to the birds if we did. We have had gold fish presented to us on a few occasions, whether from pond or tank we have no idea!