How to make gingerbread more moist and fluffy?
The biggest issue's your facing is the lack of Gluten which helps trap the air created by the baking powder (Fluffy) and the lack of fat (Moisture). I also sure the sugar would be adding a little extra strength. However the Picture on the original recipe looks dense and dry anyway.

I'm assuming you're not using butter purposely, for much the same reason you're using honey instead of sugar.

Taking these two as a basis and working on the recipe you have provided. I think the following may work for you:

3 cups Gluten Free flour
1 cups Honey (Warmed to around 50°C)
1 Packet Instant Yeast &lt;--Previously Baking Powder
1 teaspoons ground cinnamon
1/2 teaspoons ground cloves
1/2 teaspoons ground nutmeg
2 teaspoon ground ginger &lt;--- Not one
2 eggs &lt;-- Stays at 2 instead of halving. 
1 cups milk (Warmed to around 50°C)
1/2 cup maple syrup (Warmed to around 50°C)
3 tablespoons vegetable oil &lt;-- Increases to 3 instead of 2. 


Modified version of: Tasteofhome.com Gingerbread Loaf


Mix the warm Milk, Syrup, Milk, Oil and Yeast and leave to ferment for a while (15-25min)
Mix the dry ingredients, sieving to trap some extra air.
Beat the eggs until fluffy and pale (This would be easier with a couple of teaspoons of sugar if your willing to add a tiny amount?)
Mix beaten egg to yeast mix, then add to dry mix, mixing well preferably with a dough hook if you have one. It's going to be hard to do by hand as it'll be sticky as hell. 
Add to baking dish(es) and leave to prove for an hour. Then beat it back down and leave a further 40min.
Bake as described in the original recipe. May need a little longer. 


We're aiming for more of a brioche type dough rather than a soft batter or firm bread dough here so more flour may be needed. 
The above recipe should work well. The only issue may be trapping the air, if this is the case adding 2tsp of Xanthan gum to the dry ingredients will make it work much better.
The texture will be much lighter (Fluffy), the additional Oil and Egg should increase the moisture side of things.

Hope this works well for you :), I see no reason it shouldn't.