You could get a Tikka (270, 30-06, 6.5x55, Creed) then have a Wildcat fibreglass stock installed by Corlanes or any other smith. The Wildcats are made in Canada too. That would hit your weight target & budget.
used T3 6.5 swede $700
Wildcat stock $500
barrel cut to 20”, fluted and cerakote $450
bare rifle is a hair over 5 1/2 lbs, 6 1/2 loaded and topped with a z3 3-9x36.
It was the cheapest way I found of getting a light weight, that I knew would shoot well (in Canada)
A little edit, finishing a wildcat is pretty easy peasy, a bit of YouTube and reading, you’d be well on your way. Happy to help with input if you go that route.
The tikka t3 or t3x are pretty close to the weight limit, figure 6 1/2 lbs for most standard calibers, always the option as mentioned above to change the stock or even shorten the barrel. They are quality rifles not sure the price used in Canada but here 500 would be ok, if not close to new, simple lightweight scope 15 ounces and as stoxk about 7 1/2
My next rifle I add will be a lightweight. Im going to start with a lightweight. I had thought about trying to lighten up a rifle I already have but scrapped that idea.