Firstly the pi 4 doesn’t run nearly as hot as it once did, thanks to firmware updates by Rpi themselves, so a fan may not be essential, depending how much you stress the CPU ,your pi isn’t enclosed in a case. If it does overheat it won’t burn out, it throttles as a precaution. My guess you will not need a fan.
If you install the fanshim without installing the software it will run constantly. You could put a pin extension on, to attach the breakout strip , but why would you want to? The “shim” part is not for holding the fan in place, it runs code via the GPIO pins and bypasses the need for bunches of jumpers.
The reason it is £10 and people buy it, is because it doesn’t just continuously run like any cheap £1 Chinese eBay fan would. Take off the shim and you have a £10 fan.
Your kit, as cool as it looks, occupies the gpio pins via the boards breakout cable. Who knows what doubling up on the pins will do with regard to the rest of the kit? If you find you do need a fan to cool it, Why not fit a cheap fan with standoffs and run via usb or with jumpers from the breadboard on the kit (that’s a project for the kids in itself) you only need a live and earth connection to run a fan