Stripping the whole chair is the way to go. You will need a proper dichloromethane based stripper like Paramose, number 2 wire wool and a pair of marigolds. The stripper will do most of the work dab it on generously and leave it to bubble up and soften the lacquer then wipe/scrub off with the wire wool.
Im not sure where you got the idea Beech is tricky to finish, its easy peesy as long as you don't sand it too finely, 240 grit will be just right. Beech takes a stain very well but also looks good without a stain. You could finish it in a number of ways, just one or two coats of sanding sealer followed by an application of wax will give a good finish. You cold also give it a couple of coats of oil either Danish or hardwax or a couple of coats of varnish.
I notice you are in London, if you are in the North you have W.S Jenkins in Tottenham or if in the South you have Mylands for your supplies.