You don't seem all that far off what I think of as our UK summer growing season. People do vary but our own sort of "guaranteed frost free" start is about May 1st. Perhaps you have a more intense sort of growing seaon from there?

Off course at home, we may start things off earlier than that for outdoor planting. Steve mentioned beans. For better or worse, Pip likes to start these off in old centres of toilet rolls. The runners and climbing French bean (we grow cobra) get started in the (unheated) greenhouse or cold frame before getting planted out.

I wonder if asparagus would be possible? If you can, the downside would be that you'd be tying up some of your limited space for something that only has a short harvesting season. The upside is that your own asparargus spears should taste like something special.