Your trays should work just fine. I've turned half of my pond into a similar "tray bog" (about 5-6cm of water) and have been growing plants there for a few years very successfully. My plants are in pond pots (i.e pots with many holes - more like baskets) and simply stand in the water. Any plants meant for pond zone 1 (0-15cm) can be planted but I would recommend using proper pond soil. The sun shouldn't be a problem.
I've never found it necessary to keep the plants in a greenhouse in the winter (it gets down to about -15C here at times). They just mostly die back in the winter but return faithfully every spring.
If you'd like some ideas for plants take a look at my website at the plants marked with a "P". Have fun and good luck!
http://www.xs4all.nl/%7Ebultman/bernadette/plants.html

Bernadette

Madeline, I use ordinary potting soil in regular clay or plastic pots to grow a variety of marginals: Cyperus, Black Taro, Equisetum, Spiral Rush, Carex, etc. I have them sitting in stoneware bowls or ceramic cachepots or in deep trays...doesn't matter as long as the pots sit in a little water. My pond is too small for all the plants I want to grow! Sun is great and they should do well in your greenhouse in winter. Mine must make do with sunny windows. josh

Many thanks to Bultman and Josh -- one more query --how do I avoid algae in the standing water?--no fish to worry about, but birds may drink it, - although they have their own bowl of clean water nearby.