Hello,
congratulations for your 403 (which chassis no.?) while I am still wrestling with many gremlins during the restoration of my own 403, as good friend and expert Bristolian Geoff Kingston knows all too well ;-))) The pages of my web site devoted to the process since 2012 are here:
My Favourite Cars-Bristol 403
FWIW, I've tried to keep everything as original (or original-looking) as possible, trying to evaluate every modification with the eye of a Concours Judge, as I am asked to be one sometimes. Thus I went for original-looking cores for the radiators (all 3 of them), but I also put in 403-1404 an alternator masked as an original dynamo and an electronic ignition nested in the original distributor. I kept the brakes, fans, carbs etc as original as possible, the exhaust was remade following the pattern of the original Burgess system but in stainless steel and so on. I'll probably fit a (hidden) brake servo booster while I would have spared myself several weeks of agony if I had fitted an electric fuel pump instead of insisting on keeping the AC pump for the sake (again!) of originality. I'd strongly recommend everybody to effect this modification: the electric pump on my 409 never gave any problem whilst the mechanical one on the 403 is a mess. And, as you are at it, please overhaul the whole braking system and fit a new pump and cylinders. It will not cost much and it will be an insurance for the future.
Originality: I think that Bristol owners are 'not lovers of Committees', as somebody wrote, and each one of us thinks with his own head and doesn't want to conform necessarily to an obtuse search for the maximum originality at all costs. Some modifications are necessary to use our cars in modern traffic and these will be more than welcome, IMHO, even if I would stop short of fitting disc brakes on my 403.
With my best wishes for a happy ending in your restoration,
Stefano