What type of flavoring should I use for a tiered rainbow cake?

I made a rainbow cake for my granddaughters birthday, I purchased different flavorings for each of the colors. Blueberry for the blue layer, strawberry for the red layer, and so on. You can purchase on Amazon. In the 70’s and 80’s a lemon cake was very popular where you added a small box of lemon jello to the lemon cake mix and lemon flavoring for an extra punch! I did the same with all the different colors on the rainbow cake using correlating flavors of cake mix, jello, and extra flavoring. The cake was a big success because everyone talked about their favorite flavor although they happily ate all the flavors! I also added extra bright food coloring so the layers really popped.

Almond extract would evoke the flavor of Italian rainbow cookies (which are made with almond paste and extract and have a thin layer of raspberry jam between the layers).

This time of year, I'd do lemon and/or orange and almond, adding cardamom and citrus zest to the cake batter - all brightly flavored and on the light side -- though I realize those are not the flavors you typically prefer for cakes. ;o)

On COLOR, not much worry. Get a clear flavor extract from the baking aisle....they come in nut, fruit, rum etc flavors. Or use a clear alcohol (rum, liqueur) to flavor the cake).
On FLAVORS, rainbow cakes usually come in one base flavor for the cake (yes, vanilla, which you have ruled out) and get more flavor from the buttercream.
On the other hand, if you want to flavor each layer separately, go ahead and then use a plain buttercream to tie the flavors tother.
Third flavoring idea, balance between the cake and the buttercream common good combinations like banana and rum, chocolate and coffee, chocolate and bourbon, raspberry and peach, etc.
Good luck and please tell us what you eventually make.