Thanks everyone for the comments so far. A few additional notes:

I did a compression test before pulling the head. Result was around 160psi on 1,2,4 and 120psi on 3. As the spark plug was black on #3 and not the others, I suspected a top end problem - i.e. a burned valve or cracked seat - so I’ll be stripping the head down next. The standing oil was a surprise finding.

The S block is of unknown provenance, it’s at +20 but I have no idea how long ago that rebore was done. There was very minimal step at the top of the bore, so all I did was use a honing stone on an electric drill to de-glaze the bores when I re-ringed the pistons. Basically I just re-ground the crank and threw the old engine together with new bearing shells as an experiment to see if it would even run and hold oil pressure. Now I know it does, I can start to fix the ‘smaller’ problems.

This car does less than 1000 miles a year, and is my long term project to tinker with, so I’m in no rush now. It’s used mainly for car shows and local drives in the summer months with my 6yo daughter, and I do the engine work over the winter. No plans to sell.

According to the FHS website FAQ, it is indeed a synthetic oil. It may not be the same type of synthetic commonly available for sale - some of which are actually just modified conventional oils.

The most common "true" synthetic is PAO (poly alpha-olefin) or Group IV oil. The higher performance Mobil 1 oils and Amsoil are made of this. (Mobil 1 used to be all PAO until Castrol won a court case allowing them to use the "synthetic" term for modified conventional oil. After that, Mobil 1 started to use modified oils in their lineup.)

Much less common are ester-based oil. Redline is the best-known vendor of this type of oil. It has higher temperature resistance than PAO.

If FHS has even higher temperature resistance than an ester-based synthetic oil, then my concern would be how well tested it is on street cars. Even if you can use it in a race car doesn't mean it's great for a street car. Does it have any kind of automotive certifications?

To be honest, the scenario for usage is almost a chicanery-based one. If your car is regularly burning oil, it means it is consuming it. Pouring in $17/qt. oil regularly wouldn't seem to be cost effective.

You'd probably only want to use it for a one-time scenario where you don't want anyone to see the oil burning - which to my jaded mind, sounds like a resale situation.

Addendum:

Honing and re-ringing modern blocks with steel liners is different. The steel liner (like that you would find on Hondas and MINIS) is a harder material than the rings and the wear on the cylinder bore is not comparable to that which the "iron" A-series blocks experience.  The piston skirts collapse on the modern motors and will rock in the bore, which will cause the rings lots of problems with sealing. You can knurl used pistons to bring the skirt back up to spec size to eliminate that rock. (that cold also be a contributing factor to the A-series re-ring smoking-- collapsed skirts and pistons rock)

Some of the metro A+ style blocks have steel liners-- I've had success twice on lemons motors just honing and re-ringing them, but they still take a race or two to ease up on the smoking (probably due to the hone finish being sub-par).

As for no-smoke oil, it's probably a combination of thicker weight AND a formulation that when burned doesn't smoke as heavily as conventional oil, likely a synthetic. 

When you re-ringed the old pistons, did you have the bores measured for oval and taper? 

What I've found is that these A-series cast blocks do not take kindly to simply re-ringing if they have any sort of mileage on them. I know people will say, "But I used a ridge reamer and cut the ridge out of the block and it was smooth, then I honed it and blah blah blah..."  Worse still is "I honed out the ridge". Ugh. That means you just opened or bellmouthed the top of each bore and increased the bore taper.

Truth is the tops of the bores right where the ring hits and just below it are usually worn oval front to back (or distributer to exhaust) being larger than side to side (clutch to fan). Also, as you measure down the bore to the bottom of the stroke, you will get more round and less wear. The cylinder is eventually worn to an oval cone shaped, wider at the top and oblong, and rounder and smaller diameter at the bottom. Simply putting new rings on it just won't handle that. 

You are asking the rings, which are round, to seal up an oval shape at the top and also expand and contract to different diameters at thousands of times per minute. The ring material is harder than the block bore material as well (at least the oil scraper rings are). It would take thousands and thousands of miles to get the rings and bores to start to match themselves more closely and sealing things up as well as the original stuff was sealing.

Boring your cylinders with a proper boring bar gets your cylinders squared up and the final hone is only a few thousandths and gives your bores the final sizing and cylinder finish to get the rings to bed in. If you use a hone to try and go 10thou or more then there is a strong likelihood that your final product will be tapered (usually at the top).

Oil coming in and resting in your piston dish is most frequently leaking down from the valve guides. This is easy to tell if most of your smoke comes from the tailpipe after you've been sitting idling for a while and start to pull away and there's a cloud of smoke (like sitting at a traffic light or stopping at a stop sign). Traditional thinking is if most of your smoke is while driving at high speeds and steady rpms, then it is predominantly from your rings. 

For others reading this in the future, if you suspect the smoke could be from the valve guides / head problems, you can pull out your spark plugs after your car has sat overnight after a normal drive and shine a flashlight in each bore. Rotate the motor by hand looking into each one at the tops of the pistons and see if there is oil pooled on top or if it is shiny in there. You can even go so far as to twist up a blue shop paper towel and stick it in there and see if you can wick out any oil. If you find oil, THAT's a result of the oil leaking down out of your valve train and down your valve guides.

As Doug said you should have done those tests before removing the head.
I would guess you have a ring problem at this point.

If you had not removed the head I would have suggested that your next step would be to perform compression and leak-down tests.

Summer season is over, so finally got around to pulling the head. Can definitely see that the FHS oil is not burning away, as it’s all sitting there right on top of the #3 piston!  The big question is, how did it get there... valve guide seals or rings?

The pistons were re-ringed and correctly gapped to the bores less than 800 miles ago, and the head was a refurb from Mini Mania at the same time.

I had never heard of the stuff until your post.  Googling for it I immediately was directed to Amazon.  That is some very expensive oil.

Do the containers not say anything about the grade or which standards the oil meets?

EDIT:  Thinking about this more, the product says that it is not an additive but a special oil formulation.  It mentions that leaving old, traditional oil behind will allow the "contaminated" FHS oil to continue to smoke.  This oil is an expensive way of masking a problem and putting off necessary repairs.  

If the oil has eliminated the smoke you were concerned about and meets basic oil requirements it will buy you some time to drive and enjoy your car.  However, use that time to save enough money to address the underlying problem with an engine rebuild.

Is anyone using the FHS "smokeless" high temperature oil in their classic Mini ?

I did an oil and filter change today and decided to use this, as I have problems with smoke due to low compression on cylinder 3 and I don't have the time to pull the engine and rebuild it right now.

It seems to work great, went from embarassingly smoky at a car show last week, to crystal clean this week.

It's supposedly 50wt but I don't know is shear characteristics so not sure what effect it will have on gearbox/final drive life for the unique Mini design.