The test you want is at Ward Lab- it's called the "household minerals test" and costs $26.00. It will have everything you need for brewing.
Their prices went up recently, it's $27.50 for either Household (W-5) or Brewer's test (W-5a) now. So send in your water for a Brewer's test.
Don't order the KIT, they'll send you a "collection vial" and a return box for $15 more.
Now realize that a test is a snapshot. If your domestic water supply composition varies widely, many do with the seasons, that single test may not be that useful.
I'm surprised your water company wasn't more helpful. They DO have the numbers you're looking for. It's just to get to talk to the right person to give them to you. I talked to mine and got the whole picture. He also set a few facts straight on where our water actually comes from. The public report that's being mailed every year is highly inaccurate when it comes to actual sourcing. Luckily our water comes from deep local wells, is great quality, soft, and consistent. It only needs Campden treatment for the chlorine.
That said, and with Ward's recent price hike, those Lamotte water testing kits for $100 suddenly look a lot more appealing (~50 tests). That is if your water is good enough to use for brewing and can be tweaked.
Now depending on how much and how often you brew, RO water from the machines can be a great alternative at $0.39 a gallon.