I too have had a few heavy barrels, the naked weight is not the issue and a bit irrelevant. Its the final weight and all the extra stuff that we like to stick on them that important. So do you intend to put a mod, bipod, and what scope and ring set up? Get approx weight of all this stuff and add it on or better still subtract it off the the total weight you would like to end up at.
Most people end up putting a higher mag scope on these varmint style rigs e.g a night force is nearly a kg.
I have a sako 75 heavy stainless , it weighs a ton , I have tried it with a Mcmillan stock, chopped 4 inches off the barrel OAL. It's still to heavy and I will ultimately re barrel it.
The T3 varmint will be a bit lighter than my sako but not by much. If your stalking requires a bit of a hike , I would pick a suitable sporter profile . The T3 lite stainless synthetic, takes some beating or maybe look at the Tikka CTR but might be a bit caliber specific. Lastly if you are going for a sporter T3 check that your chosen caliber likes a 20 inch barrel which is how they seem to come now -Daft!!