John Deere 180 with FC540V having starting problems after about 30 minutes of use

I've got an old John Deere 180 Tractor Mower with an FC540V engine that has seen better days. I think it was built in the mid 80s. I've recently started having starting issues. First off, it seems to take a bit longer to start up. It's getting gas, but just turns over and over for a while before it seems to want to spark.

I've replaced the carburetor, Fuel Filter, fuel line, spark plug with genuine NGK and even the starter. Just today, I replaced the Ignition coil and that little silver box thingy that's about 1 inch square and 1/4" thick with a single wire that hooks to it.

I can get it to start after it turns over a few times, but after I've mowed most of my lawn and turn it off to dump the grass clippings, I try to start it and it won't even turn over. Sometimes, it will click like the starter wants to engage, but it won't turn over. It acts like the battery is dead, but I've had the battery tested at both AutoZone and O'Reilleys and they both said it tested good with all the CCA that the case says it should have.

I do put this on a trickle charger when I'm not mowing the lawn. I wonder if the battery isn't getting charged while the lawn is being mowed?

Any ideas as to what could be causing this problem?