Speak to the highways department of the local authority. At HTN we had to close about a dozen or so footpaths. They’ll tell you how to do it, but the enforcement notice will cost you. IIRC Bury Council wanted about £300 for a 6 hour closure, but it varies. Bolton wanted over £1000 to close 400m!
We had to post the closure notices at either end of the closed section 7 days prior. Also we had to marshal the footpaths that were at the extremities of the course. Some dog walkers got a bit shirty with us one year, so next time we gave that job to Ton (formerly of this parish) and he had no bother. You can’t stop anyone walking down a closed footpath, but you have to advise them that it is closed and that there is a risk of them getting hurt or causing an incident.
You won’t be allowed on Bridal Ways if it is a competitive event. Either find another route, run parallel to them on the grass, or make sure that in your literature that there is no mention of it being a competitive event ;-). The last two solutions were recommended to us by the Highways Department.
The Highways Department will probably want to see a risk assessment and proof of your public liability insurance. I think HTN was covered for up to £10,000,000 per incident!
Good luck, but expect a depressing amount of paperwork and having to deal with people who will more than likely not give a shit and for who saying NO is a much easier option than saying YES.
There is only one reason why HTN stopped. The Local Authority.