Has anyone tried Mel Bartholomew's Square Foot Gardening?

Yes I have and it's great. I don't follow exactly with the book. I've had to adjust a few things based on personal preferences, materials available, types of plants, and the use of other other methods (such as Lasagna Gardening which is very similar). I highly recommend it. My mother uses this method also and her veggie garden is amazing.

I've made the square-foot method work, but I'm in a much less arid part of the country (NJ) than you are. Are you using the "new revised" method which doesn't use very deep soil? Those can dry out very fast. Try going to a deeper bed, with a good thick layer of mulch on top to prevent water loss. A drip irrigation system would also help if you can manage one. For the best compost, make your own - the more variety in the material you put in the compost bin, the more nutritious the resulting compost will be. Even if you don't make enough and have to supplement with purchased compost, whatever homemade you can make will be an improvement. Also, if you have trees on your property, save the needles and leaves for mulch or making compost or leaf mold. For the mice, I suggest a cat. Or get some used kitty litter from a friend who has a cat and put it around the garden. The biodegradable kind is best.

I've never followed it completely, but I use a lot of his principles, and I've grown a lot of veg in pots before so I can certainly believe they'll grow in 6" if they're given the right soil and care. Go for it!

Can't help with vermiculite though as I'm in the UK. They do it in our garden centres here - have you checked?