5.8kg (12.8lb) with empty magazine, no scope, 24” barrel, no muzzle brake"
add a Pic rail to those 12.8 lbs., rings, scope, another Pic rail or two for bipod, or monopod, or both, maybe an ACI, put some rounds in that empty mag, add a muzzle brake, or a can, an anti-cant device, maybe a barricade stop, a sling - now you don't need a gym membership anymore. See it actually saves you money.
Tikka CTR in 6.5 Creedmoor is still the right answer to your question I believe. Buy whatever chassis you want and you're set. Way under budget, with top-notch performance out to 1,300-1,400 yards with low recoil for less than $1.50 a round. There's just not much downside to it.
I've run mine in a couple of informal matches last year and smoked guys with much more expensive guns. The rifle won't hold you back for sure.