As with running any motor in any class, better batteries equal better performance. I would imagine this would be very evident in a class using the Tamiya spec motors.
The best way to get a power advantage (in any class, with any motor) is having good batteries and a well-maintained motor. Since you cannot take the Tamiya motor apart (I assume?), it would be wise to stock up on a few new ones that you can switch to when you really need that extra edge. I would have the new motors dynoed, or test them on the track to make sure they all perform well and possibly find the best one of the bunch.
If you are looking for the highest advantage, I would simply get the best batteries you can afford. You may try buying one or two, just to see how much of a difference they make compared to the ones you are using now. Make sure you take care of them properly - a discharger, equalizing tray, and a quality charger/power supply should be used when you are dealing with expensive batteries.
If you are interested in our cells, you CAN order them. Unfortunately the uber-fancy website and graphics are still in the works, but we are starting to have cells available now. We carry IB3800's, GP3700's, and GP3300's. We do not use the CE Turbomatcher to "match" the cells. We have our own designed-and-built machines, and we match cells on different parameters compared to what has become the "industry standard". In addition, we have an exclusive cell-enhancing process which was also designed in-house. I know you have heard "Exclusive Cell Enhancing Process!!" before, but the results on our cells speak for themselves. We already have testimonials from various racers (even BOAT racers) which only confirm our own testing and results.
Anyway, I hope some of this info was helpful to you. You'll have the fastest Mini on the track in no time.