What do you need as a prime contractor to protect you from a subcontractor claim effecting your insurance?

Uninsured Sub Contractors.The Industry standard way a General Contractor protects themselves from the actions of A sub is, "Not To Hire",, "Uninsured contractors".

Most Reputable General Contractors will require any Subs bidding on a project to submit proof of insurance prior to bidding on the job.

So basically if the Sub has no proof of Insurance then he is not eligible even to place a bid.

If you've already chosen to hire uninsured subs and a claim is pending then you as the General Contractor are on the hook for all damages, there is no way around it.

Probably a lawyer. This could get complicated, so it's best to have legal advice all the way through negotiation and claims.