I just like blair shoot a 260rem Tikka, factory 22" sporter barrel with a ASE utra can. Have only used it on club F class shoots, but for what is basically a hunting rifle you can still be very competitive. Also used it for a PRS style matches and the light weight barrel has never been an issue.
You're pretty much spot on with your assessment, it's possible to shoot a PRS rifle well in a F class environment, but not so good a F class rifle in PRS, a dedicated F class rifle being single shot being the first main hurdle.
With F class most folk shoot a 28" barrel or longer and as much weight as possible, whilst PRS/tactical rifle doesn't necessarily suffer from excess weight a barrel under 26" is what most go for. There is a heap of information out there in regards to PRS shooting in the US, best you do some Googling rather than me do a half arsed job at repeating it here.
Basically all you want is a heavy barrel around 22"- 26" long and magazine fed, there is a crap load of rifles out there that'll fit the bill.
If it were me I'd go for a Tikka CTR/varmint or Bergara B14 hmr, put a decent scope on a start shooting.
There are loads of chassis and aftermarket stocks that are great, but until you've started shooting and tried a few other rifles you wont know what it is you like.
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