Would a fruit cocktail flavoured cake work?
Orange and lemon cakes are classics, so there shouldn't be any problem making those layers. I'm not entirely sure how you'd go about making the others, but there should be a way to incorporate fruit puree into cake batter so that the cakes still rise and set.

Blueberry, blackberry and raspberry can all be added whole or slightly chopped into a cake (think blueberry muffins), although they do make it moister so you'd want to be careful it doesn't end up too wet. This might work for kiwi and strawberry but I've never tried it with anything like those.

You might need a bit of food colouring if you want to emphasise the visual aspects of rainbow layers.