I think this is what I'm looking for.
https://www.seekinsprecision.com/srs-rail-1.html

I haven't removed the stock to see what the inside of the channel is like and what it would take to firmly attach this rail to the stock.  Anyone used anything like this before?  Happy with it?  How did you install it?

Yes, that was the first thing I changed on this gun, then the recoil lug. I can only handle about 10-12 rounds on the bench and 25 or so shooting prone. The limbs aver made a huge improvement but it's still a 300 WM pushing 180 grain bullets almost 3000 fps.

I would if I knew it but he's the magician, I just point and shoot. I know the velocity is around 2950 fps.

Also, I had to cut down the rail to allow me to move the scope back so my eye relief was where I needed it so you may want to go with the new receiver length rail if that's going to be a problem for you.

I have the exact same set up. Started out wight the factory tikka rings but the recoil was damaging them (after 60 rounds) so I got the pica tinny rail from MT and the luepold PRW rings and have been very satisfied.

I also upgraded the recoil lug and recoil pad. It helps but it's still a 300wm.

Isn't the barrel of the CTR slightly heavier than a sporter? Wouldn't that make a difference too?

That's awesome!  I love these videos. Have seen a few Tikkas do it but like he said, none on the first shot!

I have a T3 Hunter (walnut) .300WM that I may be willing to trade.  Do you know if these are a direct swap?