Modern Vespa

At around 67 mph (indicated), I'm starting to get some out-of-balance wheel hopping from the front end that increases in intensity up to 75, where it is the most pronounced. It's more irritating than anything else.

I removed the front wheel and took it to the service department of the shop that mounted my tires several months ago to have them check it's balance out (Kawasaki/Royal Enfield dealership,) but apparently due to the Vespa wheel's offset hub and narrow center, they couldn't use their normal motorcycle wheel balance equipment on it, and now knowing that, they couldn't have balanced the tire/wheel when they mounted them in the first place.

The front wheel has one balance weight on its rim, which would have been from the factory.

Is there a 'cheap and dirty' way to get the wheel reasonably balanced while the wheel is on the bike?

I have the front end jacked up so the wheel will turn free when mounted, and would think that if I removed the brake caliper to eliminate any brake pad friction, I might be able to rotate it freely enough to
generally find the 'heavy spot' and counterweight it.

Any helpful advice on this situation?

If I was in your situation where I'd never be able to get tyres fitted AND balanced, I'd buy a Marc Parnes balancer. To get me going in the meantime, I'd do what you're suggesting. Spin that wheel 10 times, and mark each 'bottom' spot when it stops. The eyeball average of the 10 marks should be pretty near the heavy point. Then add a weight to the opposite end, and try again.

Note, neither the Marc Parnes balancer nor any other static balancing will give quite as good a result as dynamic balancing, but should be good enough in nearly every case with motorcycle tyres which aren't very wide.

https://www.marcparnes.com/Vespa_Scooter_Wheel_Balancer.htm

Simple---if in N America, use Ride On----works well with small scooter wheels/tires---I am high mileage rider and do own tire work. I think over 400 tires and tubes since 1955

I also use a Marc Parnes balancer and am happy with it. (It's one part of the changes to improve front end handling)

Another vote for the Marc Parnes tire balancer. Never liked the idea of putting stuff inside a tire to balance it.