When I first tried to delete these files, one of the files was 644 and the other was 444 and I could not delete either, which is to be expected of 444. So, I, just on a whim, decided to set them both to 777 and see if that made a difference. It didn't and I didn't expect it to since, as you said, 644 should have worked.
As to your question about whether FileZilla really was allowed to change the permissions, the changes were seen by the LPCP FileManager.
Also you say something about going into the control panel to change the permissions but I could not find such a place in LPCP. I have another site that uses cPanel and I know you can change file permissions there.
So I am left with, there is something else preventing me from deleting these particular files, but I have no idea what that might be.