I don't think you'll find #3 AWG cable.
I have not measured the cables but I know that you're correct... it's somewhere between a #4 and a #2 cable.
The maximum current rating of a cable is based on its jacket material and the size of the copper conductor. Bigger cables aren't always mandated. In the case of cars like the Mini, the main battery cable only carries high-current during starter motor operation... which is very short in duration. Therefore, while a #2 cable would be great, a #4 AWG cable will work without issue.
I suggest you measure the jacket OD of your car's main battery feed cable. Using that information go to a local welding supply company and buy about 12 feet of a welding cable having a similar (close) OD. If they have a #2 cable with a jacket OD close to what's on your car... buy it. If not, get the #4 and know that it will also do the job. If no welding supply stores are in your area, visit
www.mcmaster.com and search for "welding cable". Select the largest cable with a jacket OD close to what's on your car. Again, if a #2 will fit use it, if not, get the closest #4. McMaster will sell to anyone with a credit card and the shipping has been reasonable. Browse for just about anything else you may need and add it to your order. They sell virtually everything.
All of this information assumes you really MUST replace your cable. There is an alternative. If the cable is not burned through, it is possible to repair it. Carefully cut away the burned cable jacket trying not to disturb the copper. If you have an electronics supply store near you , buy a length of large ID heat shrink tubing. You'll need a piece large enough to go over the ring terminal on the solenoid end of the main cable. Once you've removed the burned jacket material slip the heat shrink in place and use a heat gun or gentle flame from a propane torch to shrink the tubing in place. If you don't have access to heat shrink tubing, buy a fresh roll of quality electrical tape from the hardware store. While you're there, also buy a can of the brush-on liquid vinyl. Sometimes this is even marked as "liquid electrical tape". Degrease the cable and wrap the burned area with several layers of the electrical tape. Build up the tape thickness to more or less restore the OD of the cable. Now apply a layer of the liquid vinyl over the electrical tape including at least an inch of the un-damaged cable jacket. Once that application dries, apply a second coating. In the end you'll have a water tight repair of the jacket.
Regardless of whether you repair or replace the main cable, make sure you anchor it securely in a manner that prevents the cable from contacting the exhaust. "P" clips are good for this but you may need to add some additional anchor holes to the subframe for mounting points.