A neck strap is counter productive for a shoulder dropper - you need to take the precaution of a strong lower leg and a core of steel, which is easier to say than achieve. A good saddle with well placed blocks (that fits you well) can help to compensate for the required strength / balance, but that's unlikely to be an option with an RS horse.
To be honest, if the horse has form for this sort of move, and it wasn't a one off, I'd think it's not really appropriate for teaching people who aren't fully competent and riding fit. I have a shoulder-dropper in my herd and I'm very careful about the circumstance in which I allow anyone to ride him, which is a shame as he's a brilliant horse otherwise and he's not at all naughty, he's just sharp, sensitive and not very brave about life.
Denny Emerson (US 3DE / trainer / veteran of horsemanship) reckons that riding nerves primarily come from people lacking the strength or fitness to deal with what they're likely to encounter. In your shoes, that would be something you can work on off the horse, between lessons, either at by yourself or via one of the many rider fitness courses available in person and online.
During lessons, if the horse is being a twit or you're feeling insecure, I'd focus on keeping your shoulder behind your lower leg - not excessively, but just enough to give you stability in that sort of move. Focusing on a simple mantra (e.g. "heels down, leg forward, distract pony") can help to control your nerves and your thought processes while you're riding. It's much better to think something positive than "don't fall off"!