I'd personally go 3+ on the bbl contour based on a bad experience with a #2 and several well known Smith's advice to stick with 3+. Sure, smaller can work and does for many, it's just hard to shake a bad experience. You can easily make your weight goal with a 3 or even a 4. You'll hear bad "balance" claims, but that hasn't been my experience(I don't call rigs that balance at the front action screw "bad"). Anywho, I built a 257Wby that weighed 8 with the following components: Borden Timberline, Fluted Rock #4 at 25", Rem metal, McM Hunter in edge fill(26oz), TLWs, Swaro Z5 3-18x44. That's quite a bit more than your budget, just trying to show a 2 is not needed for 8lbs.
Just about to send parts to Lee for another 257wby and anticipate it being a bit over 8 due to optic choice and lack of fluting: Blueprinted PTG SS 700, PTG one piece bolt, .187 PTG lug, Rem metal, Rem trigger, Rock 4 at 25", BC Ak Ti, TLWs, HD5 3-15x42. You could go #3 and your Leupie choice to bring that below 8lb if needed. That build is well less than your budget.
I wouldn't worry about fancy triggers for "2.5-3lbs," the Rem triggers will do that with ease. I've never had one that wouldn't and I have a decent supply
I can see your wants for the cart, there's just something simple about 100s at 3700 for 400 and in point and shoot capability.
Good Luck with your new rig.
