Levi001.
I also have a Tikka T3X with a 1-8 twist barrel.
What you're going to find out about the rifle, if you haven't already, is that it won't be "fussy" about bullet weights from 45-80 grains.
I've shot all of those weights through my rifle and just about all of them shoot into sub 1/2 moa groups as far out as I get to shoot, which is 600 meters at a wilderness area close by.
I shoot off a portable, three leg bench, with front rest and rear bags.
The only bullets I had stabilization issues with was the Berger VLD 223, 90 grain bullets
In truth, I accuracy with bullets heavier and longer than VLD 80 grain bullets always has "flyers"
My particular rifle shoots the Hornady, Speer and Berger bullets with VLD design, 80 grain and under very reliably and consistently.
I have a friend with a rifle chambered in 223Rem but he has a 1-7 twist rate barrel and it shoots the 90 grain VLD/ELD bullets very well.
My rifle does have a preference for two Sierra MK offerings, #9377 HPBT, 77grains and #9390 HPBT 80 grains. It will shoot both of these bullets consistently into 3/8 moa out to 600 meters.
These two bullets need to be loaded "HOT" to perform at their best. There are three powders that do the job well and I would suggest a starting weight of 24.0 grains over CCI450 magnum or BR primers. Then work up your load in 1/10 grain increments to 24.5 grains, which should put your pressures into the 50K CUP range. The rifles actions are strong enough for more, but your cases may not be.
Now, cases are also a concern. I'm using LAPUA cases for this rifle. If you use something else, you should start your charges around 23.5 grains. Pressures build up very fast in small cartridges and the weight of the cases makes a big difference.
The powders which work best for me are BLC-2, CFE223, and W748.