In spite of its name, the Stronglight crank is not very light. I have one, and it's a nice enough part, but not what I'd use if I were trying to save weight wherever possible.
I would definitely look for a reasonably priced hollow spindle crank, like GXP, Hollowtech, etc. I'm actually not sure what all your options are for FG/SS in this style. I know there's a hollow spindle version of the Sugino 75 that looks really nice, is sure to be super strong, but costs buku bucks. (Probably half of your entire budget,
AJ1515 , based on what you said you're willing to pay for a frame.)
The SRAM S-300 is very reasonable. Lighter than most any square taper track crank (for total weight of the crank arms and bottom bracket), strong enough for what 99% of riders will ever need, and not super expensive. It's the GXP standard, which stands for Giga-X Pipe, just SRAM's version of external bearing/hollow spindle. They are not hard to install, although you need a special tool for the bearing cups. I have this crank on one of my bikes and like it a lot. One downside for some guys is that it's a 130mm BCD. If you're set on getting a 144mm, this could be a deal breaker, but if weight is a concern, you shouldn't care--in fact, you should probably be looking for 130 cranks.
Edit: I just went back up and read that you're budgeting around $2k for the whole project. In that case, you
might be able to squeeze in that super nice Sugino 75DD (Direct Drive). It's a beautiful crank, as well as being stiff and light. If I were undertaking a bike build like yours, I would prioritize the frame, wheels, and crank/BB, in that order. Everything else is just details, with multiple options available at reasonable cost and weight.