The barrel is a very important consideration. All of my competition rifles wear 30" Bartlein 5R barrels. I basically have 3 x .308s and 2 x .223s that I can shoot depending on the importance and how I feel about an upcoming match. I like to shoot them all (i.e. no safe queens), but some are set up to be more competitive than others, with varying freebore length (i.e. optimized for different bullets), twist rate, and contour.
I would suggest a 30" finish length, in either MTU or Heavy Varmint contour (if you can make weight), 5R, 10-twist, .300"/.308". If you need a lighter barrel contour to make weight, I'd look at one of the Palma contours next, starting with Heavy, or possibly M24. I started using the MTU contour years ago as an alternative to Heavy Varmint, which is sometimes enough to put an F-TR rifle over the limit in a 30" length. I like the extra weight the MTU contour places at the muzzle, but that's a personal thing; not everyone will favor a slightly front-heavy rifle.
The object is to go with the maximum barrel weight you can that will put you safely under the 8.25 kg (18.18 lb) limit. Obviously, you also want to select the contour that provides the proper balance you like with your specific setup. A 10-twist barrel will allow you to spin any lead core bullet up to ~200 gr and achieve the full BC (Sg >/= 1.5), without too grossly over-spinning anything in the 185 to 200 gr range. I would have the rifle chambered with the 2013 (2015?) US F-TR reamer that cuts 0.170" freebore. That will allow you to load 185s and 200s without any issues and lots of F-TR shooters have been very successful using barrels so chambered.
As to barrel manufacturer, my clear preference is Bartlein. To date, I have always used Bartlein and been very satisfied. This is largely because the Bartlein barrels I have used have always shot very well; I have never had a "bad" one. So, I am not generally inclined to change something that is working. However, I would also consider using a Krieger or Brux barrel without hesitation. Any of these manufacturers can make an outstanding barrel. There may also be a few other manufacturers worth consideration, but I am not as familiar with their work.
So, the rifle needs to make weight, it needs to balance in a way you find comfortable and can shoot well, and it needs a barrel suitable for the bullets you intend to shoot. In order to be competitive, this pretty much means 185 gr bullets at a bare minimum, but more likely 200 gr, or even up to 215 gr bullets. With a 29" or 28" finish barrel length, you can obviously still load and shoot any of the above bullets, but as the bullets get heavier, you will lose a significant amount of muzzle velocity, anyhow. The shorter barrel exacerbates this issue. There are those that have reported running very heavy contour barrels for F-TR (maybe 1.25" straight contour?), that have gone with a ~28" finish length to make weight. As an example, with the 200.20X bullets they are getting something like 2600-2610 fps in tuned loads, whereas with a 30" barrel, the velocity would be more like 2640-2660 fps. To be sure, ~50 fps isn't a HUGE difference in performance, but's it's not zero, either. I personally wouldn't choose to give up that much velocity for a very little extra barrel stiffness, but maybe that's just me. On the other hand, there are a few F-TR shooters that have gone with 32" barrels (or longer). Obviously, the added weight is a consideration and might make choosing a slightly lighter contour a necessity. Of course, there are certainly other ways to lighten a rifle, and in the end, you have to make whatever components you choose work together in the finished product. My personal take is that a 30" barrel in MTU or HV contour is the "sweet spot", others may have a different preference.
