I wasn't even going to answer that question.

Why??

Heat does not reduce or eliminate humidity.

Like hell it doesn't in a small space like that. I've done it for 25 years and never had one rust.

45 years as that's how long we've had the place in the family

Increased heat can reduce relative humidity because cooler air can hold more water than warmer air. But heated air can release more water vapor which can cause problems. There’s a lot of moving parts that smart people can explain better than me.

In short, there are a whole lot of other factors that play into relative humidity besides raw temperature - many of which have been mentioned.

The only safe ways to reliably control humidity are to eliminate temperature swings and remove the water vapor from the air.

Any ideas about what wood would be good for the walls? There's a lot of cedar available out here. I assume there is probably a wood I could use to cut down on moisture in the air.

My daddy came home from the big war. We lived on a cotton farm and the bool weevils came. We didn't have much money for food. He went to a VA Ag class and got $120 a month. They all made a cedar chests. I think my sister still has it with some of her most treasured possessions after 70 years.

In small spaces, all the heat does is create convection and speed evaporation of available moisture. It does not reduce or remove any moisture in the air. It only causes the moisture to further vaporize but it has not gone anywhere. Heat creates more space for the air to hold even more moisture.
A heat bar in a safe creates a stable temperature and convective currents that helps to raise the dew point and reduce the chance of moisture coalescing on a surface.

When you open the safe, you are exchanging air and reducing temps that can cause the moisture to build on the surface of metal. When you close the safe, the heat bar then does it’s job again and creates the currents that cause the moisture to evaporate before rust can begin.

In a room you can achieve similar effects with fans and a heat source but temp swings can be dramatic and cause more issues because their is little to no air exchange.

To reduce humidity it must be removed and this can be done by various mechanisms. Heat alone, ie... a light bulb, is not one of those mechanisms.

Cedar is good for moths, not moisture.

Humidity control is a complex answer. Everyone has offered valid solutions based on thier applications. The goal is lower dew point or raise temperature on metal above dew point. When this goal is met, there will be no condensation on metal.

Currently, the dew point outdoors is 65 degrees. Tonights low will be mid 70s. There will be no dew on your windshields in the morning. If the low drops below 65, there will be condensation in the form of dew on exposed surfaces.

NG has an insulated room, apparantly large enough to contain a pool table. He chose a mechanical dehumidifier and climate control. This lowers dew point in the room. Climate control allows the dehumidifier to be effective.

Pegas is using commercial safe heaters. This keeps metal above dew point. There is some air movement by convection.

64 is using 200 watts, or 200 x 3.4 BTUs, to keep a space above dew point. This is the same physics as safe heaters.

Also, consider this, each of these locations are in different climates. We do live in Texas.

There was even a Thermoelectric dehumidifier in a climate controlled lab.

All good solutions for different applications.

J.P. needs could be met by a small thermoelectric dehumidifier. It would lower relative humidity and add heat to closet.
If building is insulated and is not subject to climate swings and door openings are kept to minimum, this may be a solution. The dew point is what is critical.

I think a brief summary of everything so far would be the minimal temperature variance. You are about 30 miles due east of me and I don't want anything in an unconditioned space. However, a spray foamed area the size you are talking about should have a minimal temp change.

Is your "shop" attached to the home? If so, a cheap easy 4" flex duct, both output and input, from your HVAC system, along with insulation, would do the trick. The square footage mentioned isn't any significant reduction of AC for the rest of the home.

What about a small portable a/c in the room? I have an insulated shop with an insulated room built in my shop. I have a through the wall unit for air conditioning. It still sees big temp swings without the air conditioning running

No, it's a freestanding 40x60 shop. The shop will be insulated and the walls for the gun/ammo room will be insulated. I think a dehumidifier will be the way to go. The walls will be OSB, plywood or something else if there is a wood that helps reduce moisture.