This is why I always make sure I have a spare bike or five... :D

It's the height of summer and most bike shops worth their salt are busy. It's not really fair to expect your small job to trump a string of other small and big jobs. If they didn't seem busy when you dropped by at lunch maybe it was because they were having a lunch break? The real way to tell if they are busy is to look around at all the "used" bikes that have repair tags on them or at the big lineup in the back with repair tags attached.

And while it would be nice to get a sweet hand built wheel you will likely find the same issue is present and the local wheel builders are busy as well. Also if you get your wheels rebuilt what will you use to ride with while you're waiting? I'd suggest buy a second new or used set as a backup and send out your present set for a new spoke fitup.

Getting a wheel relaced might cost between $50 and $75, buying a whole new wheel wih a casette freehub (any bike with more than 7 speeds in the rear will need this) is probably about the same - but usually are machine built, often with inferior spokes... either of which is likey why the original wheel is breaking spokes...

You can upgrade the wheel to something lighter or fancier, but this is usually not the best idea on a commuter bike likely to slam into potholes with loaded panniers. Durability, in my experience, is best with a hand laced wheel with stainless spokes, a double wall aluminum rim, and a decent cup-and-cone servicable bearing hub (like Shimano Deore or LX).

What bike/wheel setup are you running now?