Hi Bennett!
I very much sympathize with you on this... I'm cursed with a strong sense of self-preservation myself, and even on horses that I know to be very quiet and predictable I hesitate to ride bareback. Like you, I'm well aware of the benefits, even necessities, of riding bareback from time to time.
I've had bareback lessons where the teacher would instruct me to fit the horse with a bareback pad; more in the interest of protecting the horse's back than coddling my nerves. The bareback pad had a sheet of foam padding, and the foam was covered by a sheet of suede. There was a handle at the "pommel," more for transportation purposes, but doubled as an oh-sh*t strap. Really good pads like the one my instructor had are pretty expensive, but depending on your situation it may be worth it.
A cheaper option for a bareback grab strap is to simply find an English stirrup leather and buckle that around the base of the horse's neck - tight enough to not slop as the horse moves normally, but definitely loose enough that it doesn't interfere with him. It won't be a firm point to grip, quite like the strap on your saddle, but it is a little more than the mane to grab for while you're building confidence.
Also, I like riding bareback in the winter/early spring - I'll just pop on in the pasture and walk/jog around for a little bit. Winter means more layers/puffy snow gear that soften the idea of a fall, as well as a layer of snow or soft and damp ground. The idea of falling into mud in my bootleg-Carhartt snowsuit always sounded more appealing than landing on hard summer dirt in a tank top and thin breeches.
Good luck!!