Late with this comment but better late than not at all.
Well at the very least you still have the open barrel sights to fall back on.
Yes indeed sighting a scope in at times is troublesome for many. {Including those who make a living doing it} My suggestion: give it one more try if you can spare a few shells (4-5) maybe. But before taking that 340 to the range. Go have the scope remounted and bore sighted first at a well stocked sporting goods store or a popular gun shop first. Tell them what you encountered at its sight in. You might be surprised to find all it takes is a different mounting system to accommodate your scope. There are adjustable base mounts available that help to relieve the many turns needed sometimes in a scope's adjustment. And to be honest the less adjustment off center in/with any scope the better. You certainly have a good rifle there for the purpose. I always like those little 340s myself. Short barreled with a man size stock on it. A good woods rifle. Good luck to yaw.
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be a gazillion clicks to count in the travel from one extreme to the opposite?
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Not the clicks, but full turns side to side.