Since you talk about a monthly water bill, I'm going to assume you're dealing with metered water rather than the unmetered irrigation water some areas have.

That makes it tougher. You're fairly late to be seeding and hope to have a successful lawn this year, especially if you're going to be constrained in terms of watering. The problem is that since KBG takes 3 weeks or so to germinate, it's going to be early June before you see any KBG sprouting. If you buy a mix of KBG, rye and tall fescue, the rye and fescue will germinate sooner, but you still need to water several times a day until the KBG germinates.

At that point, you can cut back to watering once a day for a week or two, then every other day (maybe, but since the grass is so young, you may need to water every day this summer).

If I were you, I'd do what I could to keep it looking somewhat okay this summer, and dormant seed in the late fall/early winter. The problem with a "normal" late summer/early fall planting here is that if you wait for the heat to break, by the time you get the seed down, you've run out of time for the grass to establish before first frost. I've had good results putting seed down just before the first snowfall and letting them germinate in the spring when the soil warms enough. There's enough moisture naturally at that time (and only at that time) for the seeds to germinate without irrigation.