But you can't do a "midwest" style garden.
Composting can be a great way to improve the soil, but it takes years to really get going. But if you're going to be there for years ...
And maybe you could even get "donations" from neighbors! I know one person who setup compost jars for her neighbors to get more compostables.
My wife for example likes to water for 5-10 seconds and that basically does nothing, you have to water significantly more as you need to put down 1/2" a day or more during the peak. It would take forever to do that with just a garden house and the pots you have can probably barely hold the water required to last a day. Reference https://earlywarning.usgs.gov/ssebop/modis for you area.
Also if you aren't buying compost by the truck load you probably aren't buying enough. For the first run you need like 6+ yards to fill up a couple beds, and the next year that will decompose to half and you will need to fill up again. Spend a bit more buy the good stuff (ask online whats good locally), the city waste compost is not great stuff has plastic and shit you will never get out plus it holds water poorly due to all the filler they add.
Which is kinda funny as my wife hates it and wants to move because gardening here is really hard-to-impossible, but she doesn't want to give up our great neighborhood, so we're at an impasse.