This thread makes me wonder if the problem is actually related to the storage and periodic starting of the car, or whether it was just a random happening.
Personally, I do NOT start my car every week when I winter store it. I put it on a battery tender, then start and run it once a month till it is fully warmed up. (Fresh oil and filter and full tank of gas before putting it away). So far, I've had good luck.
Having said that, you've replaced the battery, relay fuse is ok, relay is new, so time to go under the car, and check out the starter and wiring connections. Sooner or later, this problem will get solved. Good luck.
08 GT Convertible, Torch Red