I'm sorry but after reading your argument; you are scared.
Frivolous lawsuits aren't going to happen. How much will a private arbiter cost? Plus the arbiter will only be settleing contract disputes. The major corporations would not exist as they currently do. If anybody would be scared of private arbitration it would be the corporations. Secondly; so what if my naeighbor hates guns. If he/she has no hard evidence that I violated her rights, there is nothing he/she can do about it.
Due to the fact that any legal action would be farely expensive, most of the real disputes will be akin to class action lawsuits and not private sigle party lawsuits.
As far as breaching contracts go, it would depend on what the contract had in writing on what the course of action would be and what type of abitration would be used.
I highly doubt that you will have people sign contracts to step on your property. However, they also would not be able to sue you if they fell on your property.
Everything which you are afraid of happening through private arbitration -which would be that big of a market- are already happening now; and you have no ability to stop them.
So, tell me: the UN owns the Smokey Mountain Park; how did you stop that from happening -you didn't. How are you going ot return it back to the State from the UN; you can't. The UN is trying to pass in the US Congress a small arms treaty; is that going to pass -maybe not this time, but eventually it will. How can you stop that from happening with your small government; you can't. The votes are rigged and the politicians don't work for the slaves. The anti-gun people are already getting anti-gun legilslation passed -at the state level- how are you stopping this; you aren't.