I think, look at the tyres first, pressures and what make. It may not matter if they're budget, but it will help if they're all the same.
I had a Mini One, on an 05, it was fun on 15's with non rft's, in fact only 't' rated ones haha. It would move at the rear on a spirited drive on twisty roads I know quite well, but can't remember what make the tyres were.
The facelift One was a great car and although I changed mine to the MCS I'm now selling, I would have another 05 One because it was a hoot in the end.
Check the shocks, bounce on the corners and check it doesn't keep bouncing, or follow the car being driven by someone else, and especially with the rears check they aren't wobbling uncontrollably over bumps, but doubt you'll have this as they're pretty resilient.
The convertible weighs more of course, but a geometry check is a good idea, even if you can't adjust the rear, your requirement is to find errors. Have a ring around and see who does a four wheel alignment, should be 30-40 for checking and maybe that's all even with minor adjustment, or up to 80 if it takes a while. While there, they can check all the bushings and joints for you very easily.
Get a printout of before, target, and actual data, plus an explanation if you don't know. So tyres and suspension checked, and car driving straight.
Poor power..... your One won't get full throttle. It is programmed that way. A remap, certainly an Autologic one, will remedy that, and it is invisible. It will lift the car from 90hp to approx. 125, 7 more than a standard Cooper depending on condition. Crazy huh? On mine I put a performance camshaft in it aswell, and it was just funny from then on. Totally standard otherwise, well except the air filter element which was K&N.
Brakes are ok if you make it faster, as they're the same as Cooper and S anyway on that model.
There may not be a rear anti roll bar on your car, but that actually in town isn't always a bad thing, as some suspension compliance is good for traction in places other stiffer cars jump and skip.
My One, would clear 130 rather swiftly, and get to 135 and stay there. With the longer 1st and 2nd gears, and just the five gears as you have, it was a great combination, not as a silly car you understand, just that where your car may feel flat at 4k, they can be helped to get quite good, mine was just getting good from 3.8k and very eager to 5.5 from 4.5, whilst improving fuel consumption.
Just the remap will help, but the car has the same engine in principle as the Cooper S, so can do better as you might guess.