Tire Balance

Balancing is good. They will also tell you no need to rotate, but I do that too. If you search the forum, you'll find out why it's a good idea.

I do mine

Well it was like 6 or 7 years ago and i pretty sure i balanced mine too, trailers out at the storage yard at the moment.

Get them balanced, why wouldn't they need it?
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I balance mine, just be aware that many, (most?) trailer tires are lug centric and not hub-centric. So they will have to be able to balance them with the lugs and not the hub. Many car tire places don't have the adaptor to do that.

Balance them.

Your shop isn't exactly wrong. Balancing is for better handling and consistent road contact. Which isn't a big deal on a trailer. But it is still worth doing it to reduce wear on suspension components and reduce vibration in your living space. On some trailers we wouldn't care about those things, but on an RV, we do.

balance them

If it's recommended to balance car tires, I cannot for the life of me understand someone suggesting it's not needed on trailer tires.

I've had brake drums out of balance so bad that it took an additional 5.5 oz at the wheel rim to bring the rotating assembly into balance. You may improve the balance or make it worse depending on the balance and accuracy of the cheap brake drum. I balance the rotating assembly. Unless you check the entire assembly, it is just a feel good thing and gives folks a chance to talk about it.

Definitely balance