On the RJ, where half or more of our flights are fairly short, our "calculate" MTOW is landing weight + fuel burn, which of course is always less than "structural" max takeoff weight.
It can be equal to the maximum take off mass authorised. Same for the aircraft limited with higher max taxi mass and have to burn fuel at hold because they are above MTOMA.
So, if our calculated max takeoff weight is 45600, and we load up and takeoff at 45650, are we legal? One side says yes because you are not exceeding a structural limitation of the aircraft.
You are structural legal, but not necessarily performance legal. It could be performance limitation on your runway length or climb gradient or engine out gradients.
Obviously, if you did this, you would need to burn the necessary amount of fuel to be below your MLW.
For a full landing without an "overweight landing check", yes. Throwing passengers out is no longer acceptable! Although, if you carried a few parachutes and offered enough vouchers, there are some on this board who would take it!
The other side would say that no, you are not legal to do this since your "calculated" MTOW is a hard number and can not be exceeded even though it would not structurally compromise the aircraft.
If you have an MTOW which is calculated, then it may well be for performance reasons. If you take-off over this mass, by the 50kg in your example, then you would not be legal. It is possible to have calculated MTOWs below the Max Landing Mass Authorised, because of performance reasons. Of course, the max landing mass is structural and there could be a calculated lower landing mass for the airport being flown to, for performance reasons.
We don't deal with this issue anymore because our W & B is automated via ACARS now, but back in the day when we would have to do manual W & B some guys would apply theory #1 as told above.
Just because ACARS does it does not mean that it is right. I have witnessed a 767 ACARSed the zero fuel mass rather than the take off mass. Given it was doing an 11 hour flight then this got the V speeds horribly wrong, and at Vr, nothing happened. Fortunately enough runway to stop, even having sat the aircraft on its tail to try to make it fly.
If I have confused you, sorry, but I will resolve via PM if required. Now, off to look at a very good example of when the TORA/TODA was not long enough as the CVR transcript has to be commented on tonight.