It's getting hot in Texas! I've maintained my 2000 Infiniti I30t for years now with the help of lurking this forum, and for this issue I thought I'd ask for a little help getting the order right before spending any money. My usual MO is to randomly try different fixes until it works, but that can get expensive =).

Examples:
1) The first time it happened, I was waiting for groceries on a very hot day (100+, 40% humidity) and noticed the A/C was losing its cool. The engine temperature gauge was climbing quickly, and I cut the engine and rolled down the windows. I tried the A/C again once moving, it blew cold again but the engine temp started climbing again, so I switch to econ mode and limped home.
2) I recreated this at home on a weekend morning before it got really hot. I opened the hood, started the car, and turned on the A/C full blast. The radiator fans appear to be working correctly. I let the car sit for 5-10 minutes blowing cold with no issue and a steady engine temperature gauge. Then I closed the hood and the temperature started climbing as I experienced in #1.

Possible issues I've found from other threads:
1) Radiator fans not functioning (Eliminated?)
2) Radiator fluid / antifreeze / coolant needs to be replaced (This is my top pick -- this has not been maintained properly by me)
3) Radiator needs to be cleaned of debris -- I looked at the radiator fins and a/c condenser, and while there is some slight mud/dust/bug buildup on them, it doesn't seem like it much.
4) Failing a/c compressor increasing load on the system. -- I haven't noticed any noises, and the air is blowing perfectly cold when the engine isn't too hot.
5) Radiator coolant leak -- The only bodily fluid I've noticed from the car have been a slight oil leak that keeps me from parking it on our inclined driveway but is quite slow. The leak managed to rot out a control arm bushing, but isn't significant enough to register a low oil level between changes.

My question is what to check and in what order in order to minimize my cost. I don't think I would do a coolant flush myself at home, which is my top guess at this point. Based on closing the hood being enough of a load to tip the engine from normal to overheating, my guess is that fresh coolant would be much more efficient. As I said, the coolant has been neglected so far.

Thanks in advance for any advice!

I'd start with a coolant flush and thermostat swap. Give everything a good cleaning while you're doing the maintenance.

^ This. Also, clean the condenser and radiator with moderate water pressure, spraying from inside the engine bay to push all the funk outward. Lots of stores have a hose attachment wand that does a great job of this.