Hi Colin, welcome,
Sounds cold up there! Brambles are unique in that I find it best to dig out their main crown of root (not the whole lot, just the cental stem and a few inches around it, undercut four to six inches, a sharp copper spade helps!), after cutting the main stems and clearing them either to burn or shred.
I find usually that ground under brambles is quite weed free so often they are an easy weed to deal with, necessitating little further clearing or mulching. But their main, central roots, if left in, do regrow when the chief stem is left in place.
Good luck with the growing and all those tunnels!