Recoil starts the fraction of the second after the primer ignites the load.
I've tried different bullet weights and I found a heaver bullet will always print higher, than a lighter on at a faster velocity, due to the slower velocity and longer time in the barrel, thus allowing recoil/muzzle climb effect the round. I know we are talking only 1000 of a second but it stll effects the bullet since it is in the barrel.
so, again we need to define terms a little better.
The forces inside the barrel do indeed cause movement of the barrel up and down.
This has been described as harmonics, the spring like oscillation that occurs as the bullet and gasses load and unload on the barrel.
However, the previous poster is correct that recoil (as defined as the stock moving into the shooter) is not a factor due to the time delay.
tuned muzzle brake by definition, is designed to minimize the barrel whip caused by harmonics.
Heavy and fluted barrels also resist this movement.
There are dozens of factors, the key is to correctly identify the PRIMARY factor that is causing the problem.
A new stock is unlikely to change the barrel harmonics that causing the majority of the problem described. It may help in general, but it will not substantially address the root cause of the problem described.
Muzzle brake and or different profile barrel.
One or both will work.
PS, shorter barrel has the same effect also of addressing the harmonic characteristics.
You could saw off at one inch increments and see where you find a sweat spot.
You might get lucky and catch it early.