A novice can render a pair of SUs on an engine almost undrivable; three (3) SUs increases those chances by 50%.
The best recommendation is to find a local specialist that's capable of getting the tune spot on.
Tuning a Healey is not an exact science, and theoretically, anyone can do it; the reality is that it takes experience to recognize when the adjustments don't yield the desired response, and figuring out why. Air leaking past the shafts, rendering repeatable idle settings nearly impossible, is a good, and frequent example.
Don't think that a prospective tuner is trying to gouge you when he tells you that you need a complete tune up. The carbs cannot be satisfactorily adjusted unless it is verified that the valve settings are correct, the points & condenser are in good order__and properly gapped__along with the ignition timing correctly set, and a good, clean set of plugs, distributor cap, rotor and wires. Only when all of the above is in good nick can you even contemplate to mess about with the carbs.
So to directly answer your questions, yes, there are a number of us here capable of tuning a trio of SUs, but beyond using an actual SU wrench, there's not much else you need to access the jet nuts__most of the time, if they're clean and in good order, they can be turned without any tools. I am however, confused by your terminology; specifically "built in" and "fan type"__what do you mean by that?
If I walked up to your car, and was allowed ten (10) minutes to tune the carbs, and you swore that everything else was as it should be, the very first thing I'd do is to verify that the fuel level in all the float bowls came to the same height__better that they're all even, than to quibble about the level being off a millimeter either way! Next step would be to remove all three (3) dashpots to verify the pistons can rise and fall smoothly and evenly, plus to examine the jet needle for wear__too common, and might as well stop right there! Verify the pistons fall evenly into the jet well, bring all the jets to the same height (about 1-3/4 turns down from being flush with the bridge) and then have a go to see how well the linkage is synchronised.
Okay, so that might take eleven (11) minutes!