Savage B-22: if I free float the barrel do I need mess with the stock?

[twofewscrews]

As title stated I am considering free floating and pillar bedding a Savage B-22. The rifle in question is chambered in .22lr, has a heavy bull barrel, and sits in a synthetic accustock. Free floating it is actually the simply part as the channel that the barrel sits in is removable. See the attached image.
I have a couple questions.

Would I need to add any material to support the barrel if I free floated it? The part I circled in red is the area is where I was thinking of adding the additional barrel support. Would it make sense to do this to both areas where the stock screws into the receiver (I think this is called pillar bedding or something but I'm new to the game)?

Should I reinforce the stock after removing the tray that the barrel originally sat in? I could do this either by adding a barrel tray which doesn't make contact with the barrel or filling in the empty space with epoxy.

Thanks for your time

 

[Varminterror]

Hmm… so why is that “tray” in there at all? Is it fixed to the stock in some way? Bolted in place, pinned, keyed, etc?

 

[twofewscrews]

Hmm… so why is that “tray” in there at all? Is it fixed to the stock in some way? Bolted in place, pinned, keyed, etc?

Screwed into the stock with small screws. I believe it serves two purposes, it fully beds the barrel against the stock and provides additional strength to the stock.

 

[Ru4real]

Personally, I’d proceed in steps. Shoot groups for record if you haven’t already.

Finnish your pillars. Then shoot groups with the channel in (not free floated) versus the channel out (free floated) using the same ammo as your control groups. What you should do next will probably be obvious.