Anything that touches the barrel might have an impact on harmonics so by stiffening the stock we are removing those we cannot control but we can still introduce methods
we can control in order to tune that barrel.
Initially the most popular way is to simply adjust the torque on the action screws and this will impact the tune of the entire group.
There are other methods like dampening devices, extra screws, adjustable muzzle brakes (boss), etc... all from simple to more exotic.
A 20" SPS is short enough that has many accuracy nodes so I would not be too worried about it, other than adjusting torque in the action screws.
In this case I would go a bit deeper than skim bedding and to give more strength on the lateral structure that is also plastic so all the way from the front
to the back and making sure one has good contact with the pillars. But it doesn't have to be super deep neither imo. I think the most important part
is in the lug area to make sure that whatever solution we implement to stiffen the front is one solid unit and one with the front block and also is not
affected by heat. That is also why we should never allow the barrel to touch the bedding under the shank ahead of the lug.
It should be a fun project and if
@Bayourambler has a collision repair shop I am sure he has the tools and skills do a better job than most of us here.