OP has a Banish 30, so it's around 11 oz configured at 7" and 14 oz at 9" configuration. He'll need a sub 5 lbs rifle to even come close to 6.5-7.5 lbs goal. 8-9 lbs with the current suppressor and over 20 ounce scope is a more realistic goal with a 22" barrel and 26-30 oz chassis.
The reason I know this I have a 7lb suppressed .308 with 3-10X42 SHV, Mountain Tech rings, and 14 oz Omega 300 suppressor. My stock weighs barely over 1 lb and just over 3.5 lbs for the barreled action if I add in polymer tg, magazine, and action screws. The rifle alone weighs 4 lbs 10.6 oz.
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IMO a brake will tame recoil much better than a suppressor, and I can tell you that my Howa isn't fun to shoot with 180 gr or heavier bullets. 168 gr is about the upper limit that I want to shoot in any quantity, even suppressed. I can't spot impacts on most bullets even with the suppressor, as this rifle has a very fast and sharp recoil.
@Upnorth1, again I'm not trying to talk you out of your build. However have you looked at the
Weatherby 307 Alpine, it is basically the build you described but on a M700 footprint 307 Weatherby action and under 7 lbs. I don't know if you can trim the barrel back to 22" and stay out of the flutes, but I'd bet at $3200 it's cheaper than you can build. When you want a light rifle, it's almost always cheaper to buy factory.
If you go forward on the Tikka build, I'd go with your original cartridge of choice .300 WM. Id buy a SS T3x lite, drop the barreled action in a MDT Hnt26 chassis, after cutting and threading at 22". Even at this you'll be over $2500 just to assemble.
If you build a 7 PRC Tikka, MDT Hnt26 $1400 standard chassis, $1600 for the folding, $50-70 for one AI mag, $800+ for a donor action (if you don't have one), cost of barrel of your choice, plus whatever your GS charges to install, chamber, thread, and anything else you want done to the rifle. Stuff adds up quickly for sure.