Very unusual for float levels to be out on the 600, unless somebody before you tampered with them or replaced the float valves with spurious one. Did you use a clear pipe to check the levels? If so it can be easy to get it wrong. When you open the float drain to fill the pipe you may get an air bubbles and put the reading off. With a long pipe and it has filled lift it up tap it a few times and then drop it down below where you had it. This will normally dislodge any air bubbles.

Check the downloads section  I uploaded  the sizes and type or O ring to use there.  I just uploaded Information on the float bowl seals

Post pictures anyway and then we can look and see

Your plug caps will  give trouble long before the Coils ever will.Thankfully the caps can be dismantled and cleaned out. The biggest issue is the carbon resistors which can go faulty taking the resistance up to Meg Ohms which cause the caps to track on the outside and can be seen clearly at night.
I like the look of the original caps and usually buy a cheap set of NGK caps and break them open and use the resistor as it has a stainless outer wire wound resistor and will usually last as long s the coils.When dismantling the originals there is a tiny brass spacer which is usually corroded and breaks if you look at it the wrong way, but can be discarded when using the NGK resistor. The spring should fall out but may be corroded and stuck. Inside the cap there is a flat brass part which is part of the screw which needs to be cleaned usually scratched clean with a sharpened screwdriver.  Clean the spring and reassemble. Use Dielectric grease on the boots and all will be well for years.
On the carb front it is as you said most likely the pilot jets. I gave up trying to clean them and now replace them. The reason being that the internal of the pilot jet has a web type structure which is virtually impossible to clean properly.Replace the O-rings on the float valve and main jet with Viton/FKM or FFKM ones and it will save taking them apart  for a while again. Sizes are in the download section https://foc-u.co.uk/index.php?action=downloads;sa=view;down=93

Ok. obviously rewired with minimal wiring. Look at the underside of the relays and put the numbers to the relevant wires on your diagram and I will show you how it works on your diagram

Attached is a picture of the two most common relays you will get at the Motor Factors with the inner workings shown

Any BSAs I've rewired had dual filament bulbs. The charging systems didn't have enough power to light two bulbs at the same time. It struggled with one. The days of candles in brown bottles is what we referred to them as 

I presume you mean gas flowing, Smoothing of cast rough intakes ports on the carburettor engines reduces atomisation of the fuels and can lead to unburnt fuel which then reduces power. Smoothing the exhaust ports expel the burnt fuel as fast as possible is a better option. Not worth the effort on the FZS600 as the engine is in a highish state of tune as it is. You would most likely end up with less power engine. Reprofiling the cams may be a better option but is it worth the cost for a small gain.

To lower the electrical load I have fitted LED lights - made a massive difference.

Same here and I replaced the White connector with Anderson powerpoles.
Plans are to do something similar with one of the connectors under the infill panel.
The original wring and connectors are fine, but when water especially salt water gets in and corrodes the connection and you get issues, unfortunately as corrosion travels back through the wires the strands become insulated from each other creating higher resistance in the wires reducing the current flow in the wires
Fitting MosFet Rectifier/Regulator helps and will improve battery life with it's faster internal switching plus it runs much cooler than the Shunt type, but it does not solve the underlying issue, Dielectric grease stops the corrosion, but doe not fix it and the only real solution is wire replacement, which is not the easy.

Check my reply on your post https://foc-u.co.uk/index.php?topic=27916.msg333363#msg333363

With a 1.5hour wait on Monday for my wife  who was at a physio appointment, I headed for a coffee in the shopping centre and took a look at the magazines in Easons Book shop first. The magazine that caught my attention was Classic Motorcycle Mechanics because it had a picture of a yellow FZS1000S on the cover, thumbed through it and bought it. Into the cafe and settled down with my coffee to read the article.
The bike they used for the test was a yellow 20003 S model with 10,000 miles, looking like it came out of the showroom totally standard.
A good article, but didn't tell me anything I didn't all ready know and the article contributors loved it.
Definitely worth a read if you have not already done so.
 

Are you sure you didn't put plug lead 3 on plug 4 and plug lead 4 on plug 3. Very easily done and looks right, especially if the leads on the tags are missing

Down load the manuals here https://foc-u.co.uk/index.php?action=downloads;cat=2
Highly unlikely your tensioner is giving trouble but it is far more than likely to be a worn cam chain or the tensioner has not taken up the slack yet. Common for engines with cam chains to rattle before the tensioner takes up the slack  because chains do not wear evenly. Only time I ever saw an stuck Fazer 600 tensioner was on a bike that was out in all weather and not used for 4 months under a tarpaulin not to far from the coast and the tensioner was rusty, as was the rest of it 
You only need to remove the tank or prop it up securly. Pull up the plug caps carefullyRemove all the cam cover screws. Remove the Rotor cover and tie it up out of the way.rotate the Rotor anticlockwise until the 'T' mark lines with the joint between the crankcases at the front of the engine.See picture
Lift the cam cover and check the timing marks see picture of the actual cam and cap marksmark.
If the cam does not line up close to the cap then rotate the rotor a full 360degreeds and line up the rotor marks again.
If the cam marks are less than 5mm off to the rear then the rattle is probably from the tensioner not having taking up the chain slack yet.If 5 to 10mm off to the rear chain is wearing and tensioner has not having taken up the slack yet.More the 10mm off to the rear, the chain needs to be replaced as the tensioner is probably at the end of its adjustment.Might be worth checking the tappats as you have the cover lifted.

When you are putting everything back, put ordinary garage grease not silicone on the cam cover gasket and make sure it is back on properly.You can be liberal with ordinary grease as it the excess melts off