Do I need a corded drill to mix thinset?
First of all, I have to say that with the volume of work you have you can just mix the thinset by hand. I once had to install about 2.7 square meters of tiling and mixed several small batches of thinset with a spattle - no problems at all, I wasted much more time on applying the thinset and installing the tiles evenly. Totally not worth risking the drill.

Next, you should not use any drill for mixing unless its manual says it's suitable for mixing or that it has overload protection that slows the drill down in case of overload. Drills explicitly suitable for mixing are typically professional drills with very low RPM (500-700 RPM) and so very high torque. The "HF corded drill" you reference is nothing near that. Drills with overload protection are typically professional cordless drills with smart electronics that monitors all the parts for overheating and reduces the output power in case of overload. The Milwaukee M12 you have likely has the right type of overload protection (that's the impression I have from reading the online specs of the drill) and so you likely could use it for this purpose.