With those wall anchor fittings - when done correctly - I understand that a washer would be needed as else they will sink through the widened hole in the plaster? Wouldn't that washer then show as it will be wider than the bracket I suspect?In my opinion they are significantly far enough from each other for you to be able the interset cavity fittings that you have. Perhaps try to ensure that "wings" will be at 45 degrees to horizontal to ensure that one doesn't clash with the other. Or rather that one will be at 90 degrees and the other at 45 degrees- thereby avoiding any clash.
I might be missing something!