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Weight hardly affects fuel economy unless the commute is heavily stop and go traffic, and even then 40 lbs isn't going to be all that significant.
Tire plug kit, pliers, and a mini air compressor is the best option. You can make a repair in as little as 5 minutes without taking the wheel off the car. It would permanently fix about 90% of leaks. Plug kits can be purchased for $2, and inflators for maybe $15.
A tire shop wouldn't repair that screw because it's in the sidewall. Again, I'd just throw a plug in it and see how it holds up. Worst case, a leak develops and I have to replace the tire eventually.
With those type of pictures where the screw is mostly sealing the leak, I get my ream tool and tire plug ready, pull the screw, and quickly ream the puncture, and then quickly plug it. It can usually be done with minimal loss of air so that you don't have to spend an eternity filling the tire.
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