Kettering Classic Motorcycles work together with the AMC owners club and will guide you to the right bike.
Theyare not far from you at all
Personally I do not think you will do better than a late alternator equipped 350 Matchless or AJS.
1958- 9 onwards.
The AMC is one of the biggest if not the biggest one make club; the spares back up is first class as is the knowledge base,
A section meets very near you.
They are also a very friendly club,[ you will suspect I have a vested interest?] and will be delighted to arrange try outs for you.
you will get a damn good one of the above for 3k.
HOWEVER
On no account buy anything customised, that bobber chopper shite sort of stuff, nothing that is described as Trials style or green laner, or what is popularly known as an AMC "lightweight"
It may be a good idea for you to wait till the summer and get round a few rallies to see what takes your fancy?
