I know those bikes very well, I have two of them and one of them was my race bike for 23 years. But, at this time of year, time is not in abundant supply. I've got plenty of my own stuff to take care of without having to think about anyone else's stuff.
Hard starting when cold ... These bikes are really finicky about their carburetors at the best of times. If fuel was left in the bike over the winter, it is not going to be happy about it now. Was the bike like this last year, or is it just now that it's like this?
They benefit from one size richer pilot jets. They benefit from iridium spark plugs (CR9EIA-9 extended tip - the extended tip won't collide with the piston if it has stock pistons in it). The stock charging system is not good; if the battery is not up to snuff, it won't crank fast enough to start. I've got a pile of stock voltage regulators, most of them fried (mine now have modern regulators and lithium batteries). The valve clearances do indeed need to be spot-on. If the engine is tired, that won't help.
The rubber adapters from the airbox to the carbs harden up over time. If you take the airbox off (necessary for working on just about anything on these bikes), and those boots have hardened up, they will never seal again. I don't know if you can still get new ones. I've gone through a few sets of those. Usually this doesn't affect starting and idle, but it will make it lose power and stumble and miss under load.
Options:
Pro 6 Cycle might be an option. Both Craig and Mark are mechanics who date back to when those bikes were current.
HB Cycle might be an option. The owner, Phil Hodgson, is a major supporter of the VRRA (Vintage Road Racing Association) and he supports a few FZR-riding competitors.
Blackstock Cycle might be an option but I don't know anyone there. They support Kirby Crosby, another FZR-riding VRRA competitor.
Contrary to someone else's suggestion above, parts availability for these bikes is dwindling fast, and is largely behind my decision to retire my FZR race bike and start over with a new (almost) bike. (I built an R3 for this year.) My FZR can stay right where it is, in front of a wall of trophies.