I made a vegan pumpkin pie without sugar in- I used brown rice malt & maple syrup to sweeten it - for dd for the winter solstice. Seeing as it's summer where you are, you could probably do some frozen fruit / moussey type of thing. Or like a vegan type of cheesecake without a base or with a oat base with tons of fresh berries on top.
I haven't got any exact recipes to hand but you can make vegan cheesecakes with silken tofu ( organic of course ) bung in the blender with maple syrup, melted brown rice malt, purreed fruit, spices, vanilla etc. to taste If you're going to bake it, bake it in a moderate oven until set & then cool & chill & top with fresh fruit & maybe some form of soycream. You can make your own with tofu, some sweetener& lemon juice in a blender.
If you don't want to bake it, you can use agar agar flakes dissolved in hot water to set ( check out packet for qtys). In fact, you could scrap the cake idea all together & make a real fruit jelly with fruit juice, fruits pieces & use agar agar to set it.
The other way to make a vegan no sugar mousse is to use coconut cream & lemon juice & purred fruit. I use organic dried mangos soaked in a bit of water overnight. It is quite a funky tasting concoction so I'd definitely do a practise run
If you wanted a proper cake that is gluten free, that will actually taste like a cake to other people, these are generally made with ground almonds instead of flour. Not sure how you feel about nuts ( or eggs ) & a 1 yr old. I know my dd reacted to them until very recently. The idea is you whip up several egg whites until very frothy & firm. Fold in the egg yolks, sugar ( or whatever else you're using ), melted chocolate is the usual one although I had a recipe somewhere that used carrots, then fold in the ground almonds & bake in a mildish heated oven. A bit like baking a quiche.
The other possibility is to try using spelt flour as it is very low gluten ?