Alex,
A heavier plane doesn't give you more braking force available, but a heavier plane needs more braking force to be stopped.
So, (I honestly don't really remember from my schooldays) are you sure that the formula you give is not for calculating the REQUIRED braking force?
Is 'g' the decelleration rate?
So from the formula as I (mistakenly?) read it, a higher A/C weight 'M' will lead to a higher required braking force.
Or the other way around:
at a given available braking force (for example max manual braking) and at a fixed braking action 'm', then a lower weight 'M' will give you a higher decelleration rate 'g''.
Spoilers after landing, or at rejected take off, kill the lift, bring the weight of the A/C on the wheels, thus giving the wheels more grip i.e. increase braking action. You will then need less distance to stop or you can stop a heavier A/C in the same distance, or you will need less braking force to stop your plane in the same distance, thus reducing brake wear and brake energy build up in case of a short turnaround.
FYI, also in my company, independant of use uf assumed temperature or not, we always use speeds for the actual TOW.
Greetings, Ziggy