A .223 Rem with a well-developed load using one of the heavy, high BC .224" bullets in the 88 to 95 gr weight range is fully capable of winning F-TR matches, especially in midrange competitions. In the right hands, a .223 with heavies can win at 1000 yd, but that is a much tougher row to hoe. A solid .223 Rem setup with heavies can be made to shoot extremely well, and the relative lack of recoil is almost certainly a contributing factor. However, such a setup will be giving up a considerable amount of wind deflection to .308 Win rifles with 200+ gr bullets in long range matches at distances of 800 to 1000 yd. Thus, one must decide if they want the benefits of shooting a .223 Rem primarily at 600 yd (or less), knowing that they will be putting themselves at a slight disadvantage against .308 Win shooters in long range matches. The
relative amount of shooting anticipated in midrange versus long range matches would likely be a big part of that decision.
Regardless of which cartridge you choose, the key here is participation. New F-Class shooters can pick up a wealth of information simply by talking with more experienced shooters at matches. Thus, one approach would be not to go overboard buying gear initially, just putting together a minimalistic setup that will allow you to get started and participate, knowing that you will likely be adding more gear as you gain experience and figure out the direction you really want to go. Alternatively, there is nothing wrong with buying a decent setup at the outset that will allow you to compete for some time without necessarily needing any major upgrades.
Regardless of which direction (i.e. cartridge) you choose, I would suggest a 30" barrel length, and the rifle chamber/throat will obviously need to be set up for the relatively long heavy bullets favored by F-TR shooters (88-95 gr for .223 Rem, 185-200+ gr for .308 Win). As a general rule of thumb, you'll want a
minimum 7-twist barrel for .223 Rem, or a 10-twist barrel for .308 Win. There may be other considerations for a .223 Rem that are not absolutely essential
per se, but that I would strongly recommend, such as a 0.219"/0.224" bore/groove configuration barrel, which can be obtained from Bartlein or Brux. It may not seem like much of a difference, but the tighter 0.218"/0.224" barrels may increase the potential for bullet jacket failures, which is something that can be effectively cured by going with the 0.219"/0.224" configuration.
For the stock/scope/trigger/etc., personal choice is the biggest determinant here, there are many choices that can all work well, so the choices need to suit the shooter. Careful selection of these components can help prevent rifle weight issues. However, it is not that big of a deal to
add weight to the rifle where necessary, certainly it is much easier to add weight than it is to remove weight from a rifle that is over the limit for F-TR (
8.25 kg, or 18.18 lb).
Along the same line of thinking, the Tier One bipod will work; certainly it is sufficient to get started. I used a Harris and/or Atlas V8 bipod in F-TR for several years before moving to something else. It is important to note that the use of "traditional" bipods for F-TR such as the Harris/Atlas/Tier One/etc. has become less common as many F-TR shooters have gravitated toward the ski- or sled-type bipods. The discussion about the two different styles of bipods (traditional versus ski-type) can be saved for another discussion. The bottom line is that although there are different caveats associated with their use, both types can work well, but the rifle has to make weight with the chosen bipod. I mention this here because the Tier One bipod is extremely light for an F-TR bipod, which may be something that will have to be dealt with at a later date if a decision is made in the future to switch to a heavier bipod.
The primary reason for using a different kind of bipod is the width of the footprint. The ski-type bipods have an exceedingly wide footprint relative to something like the Tier One, and it can make for a more stable shooting platform by helping to minimize torque from the rifle. Obviously, this beneficial effect would be more pronounced with the heavier recoiling .308 Win than a .223 Rem. If you already own the Tier One bipod, it's not going to be a big detriment to use it to get started, and observing other F-TR shooters' setups may help you decide whether some other style of bipod is better suited down the road. Another option that I favor is the Long Range Accuracy bipod (
https://www.eurooptic.com/long-range-accuracy-f-class-bipod.aspx), which is a traditional bipod, but that has a much wider footprint than a Harris/Atlas. I am one of the few F-TR shooters still using a "traditional" bipod, but it works for me. Like many of the ski-type bipods, the LRA bipod is substantially heavier than a Harris/Atlas, and so the weight is definitely a consideration.
The good news is that you're doing the right thing by thinking about what kind of a setup you'd like and asking questions here. You can end up with a solid rifle setup that is competitive regardless of which caliber/cartridge you select, so I wouldn't be agonize too much about that decision at this point. As I mentioned above, the sooner you get started, the sooner you will be having fun and meeting other like-minded folks at matches. If at some point down the road you decide you really want to jump into the F-TR game in a big way, your experience up to that time will better enable you to either modify your setup or acquire something new if you so choose.