I run a 67 Camaro in amateur hill climbs that's maybe one step more sorted than your Mustang, so I can tell you it's absolutely doable without turning it into a race car. The guys in full builds are going for times. You're going for seat time and a fun weekend, and there's a whole class of us doing exactly that in street cars with interior and carpet still in them. Brakes are where you shoud spend your money first, not suspension. A slow climb is actually easier on brakes than a highway pull since you're rarely on them, but the section transitions matter. Your gravel-to-pavement course means you'll want pads that handle heat without fading, and fresh fluid bled right before the event. Standard upgrade route: better pads, stainless lines, fresh rotors if yours are worn. You do not need big brakes for amateur class. What kills people's days is boiled fluid, not lack of stopping power.
The gravel and pavement split sounds scarier than it is. You just drive the two sections differently. Gravel rewards smooth and patient, pavement rewards commitment. Most first-timers are way too cautious on the pavement and way too eager on the gravel. Do a couple track days or autocrosses first if you can, and walk or drive the course slowly during practice if the club allows it. Knowing what's over each crest is worth more than any part you can buy.
Cooling is the one place I'd go beyond a bigger radiator. An old V8 crawling uphill in summer heat with no airflow is a genuinely rough situation, and these old engines were never great at shedding heat to begin with. Bigger radiator, yes, but also make sure your fan setup actually moves air at low speed, because that's your real problem on a 14 percent grade. A shrouded fan that pulls hard at idle matters more than radiator size. Check your timing too, since overly advanced timing makes these old motors run hot under load.
One practical thing: look up the tech inspection sheet for your local club now, not the week before. Most amateur classes for older cars need a roll bar at minimum, decent belts in good date, and a battery cutoff or fire extinguisher depending on the club. Budget a few hundred for that stuff and you're fine. It's a wallet-friendly hobby if you resist the slippery slope, which I have failed to do for six seasons now.