My 16" Goodyear Eagle NCT run flats have printed sidewall rating of 52 lbs.

The Cooper manual states for a 195/55/R16 run 33 lbs. but doesn't clearly state "runflats".

What is the deal as the tire rating says "52 lbs. maximum" but 33-52 is quite a range? What do you recommend?

 

Check out the plate with tire pressures on your driver side door jam

For every tire there is a max inflation pressure in psi marked on the tire.
Don't worry about that for now.

For every car there is an optimal tire inflation pressure based on the load on the car.

For your MC with 16" runflats it is 30 psi front and rear tires for a load of two people and a max of 35 psi front and rear tires for a load of up to 4 people and their gear.

So as an average 33 psi is not a bad tire pressure for general use since you don't always know how many people will be in your MINI unless you carpool regularly or are going on a long trip.

The 52 psi is the max recommended pressure for that tire and the load rating (how many pounds that tire can be loaded with) will max out with that psi.

Your MINI is not heavy enough to make use of a tire pressure above 45 psi.
Even with racing autocross where psi climbs a bit closer to 40+ it is still not that high.

When in doubt always read your MINI owner's manual. The drivers door jam has the tire pressure numbers for all MINIs in various configurations. MCS gets a slightly higher psi for 2 and 4 passengers.

 

You can also "tune" the psi to ensure that you have the proper contact patch based on your MINI and your driving style. This is a old autoxer's trick. I have used two methods:

Chalk Marking Method:

Take some chalk or, better yet a tire crayon, and make a line that goes from the shoulder of the tire to about 1/2 inch into your tire tread. Do this in three places on the tire. Now drive as you normally would for a couple of days and examine the marks.

If the marks are worn off over the shoulder, you psi is too low. If the marks are untouched, your psi is too high. Ideally, the marks should be worn just to the edge of the shoulder.

Pyrometer Method:

If you have access to a pyrometer with a probe, you can use it to help tune the psi. Basically, your goal is to get the tread temperature to read the same at three points across the width of the tire tread. Measure the temperature near the inside and outside (edges) of the tread and in the middle.

If the center temerature is lower that the average of the edges, your psi is too low. If the center temperature is higher than the average of the edges, your psi is too high. Ideally, the temperature should be the same at all three points. However, unless you have adjustable shocks and camber plates, do not expect to achieve uniformity; use the averaging technique.

WARNING!

The above techniques are meant to tune the psi within acceptable inflation recommendations. In no case should you exceed the maximum psi as specified on the tire.

FWIW,

Theo