Also, great point about liability. OP, a HUGE line is if/when you take any money.... then it becomes commercial. I've heard horror stories of people taking a few bucks from a neighbor for hauling something a few miles and admitting that to a weigh station, cop, etc.. If you're just doing it for the school as a volunteer your liability is likely minimal in that regard. I'd be REALLY weary taking ANY money though. Fwiw, I'm sure the school has some type of umbrella insurance policy but you really don't want to deal with that. A few bucks for gas, etc. AFTERWARDS is probably okay but tread carefully. Transportation rules/laws are super strict.
Edit: Sorry for the wall of text, all. I have ADHD and Autism...sometimes when my brain hits a topic, it doesn't stop overflowing information. It didn't look that big on my screen, but when I looked at the post...whoa...lol.
The school's insurance covers them, not you. Volunteering adds in a whole 'nother layer of issues you really don't want to be a part of either.
There's reasons why "compensation" is the word used in the FMCSA regulations, as opposed to something like wages or income. To normal people, wages and compensation is often used interchangeably. But from a legal standpoint, compensation is receiving anything of value in exchange for services performed. Donations and gifts are also considered compensation.
Pull a trailer, and get a nifty school band t-shirt to say thanks? You've just been compensated.
Spent $23.12 on gas, and they hand you a $50 gift card to Bob's Pizza for your trouble? You've just been compensated. They hand you a nifty plaque for trailer puller of the year? You've just been compensated.
The trailer doesn't even need to be involved here either. Toss a load of instruments in the bed of the truck that aren't your kid's instruments, and you're doing the same thing. Having the trailer in tow just makes a heck of lot easier to claim commercial hauling, as the trailer will be titled/registered to the school.
Easy answer to this - call your insurance company, and ask them what they'll cover in the event of a claim related to pulling the school's trailer, and/or hauling school equipment in your truck.
OP doing this as a volunteer adds another layer of stickiness to this problem. Typically, a volunteer driving their own vehicle for a nonprofit, which almost all schools are, would be running under their personal vehicle insurance policy while volunteering. This is normal, and legal, but does not make for a good deal for the volunteer. Say something fails on the trailer, and causes you to lose control and hit another car. Your personal insurance is first in line. The other party is very likely to name both you and the school in the lawsuit aftewards. While the school may very well pay out any claims in full, this is still a nightmare you don't want to be a part of. Especially if they say your totalled truck is your insurance's problem...remember that part about a personal policy may not cover this?
While my personal opinion is one of run the hell away from this, I do completely understand the desire to want to help out. Chances are OP could do this for years, and nothing bad will ever happen. But if it does....things can go real ugly real fast.
If I found myself in the situation of really wanting to do this, ideally it would be done as a part time employee of the school driving a school owned vehicle. This then shifts the entirety of liability on to the school and their insurance program, leaving your personal insurance completely out of the mix. This also means that in the event you ever becoming injured in connection with this towing duty, you're covered under workers comp insurance. Volunteers rarely are covered. The sad truth is that in the USA, the laws protect the organization FAR more than volunteers, and far too many organizations take advangtage of this in ways that cover the organzation's proverbial ass, not the volunteers.
If you really want to use your personal vehicle, I'd still suggest doing so as a part time employee. If they insist on you doing so as a volunteer, it's not to protect you, and that's a big ol' red flag on fire. Make absolutely sure they have a provision for non owned auto coverage that includes comprehensive and collision. If they stop at liability only, your totalled vehicle is your problem. Even if your insurance does cover this use, do you really want to be on the hook for years of increased insurance premiums?