How to keep cake soft when served frozen (Ice Cream Cake)
I don’t think you can achieve room temperature softness but, I think you can start with the following considerations:

Minimize the water content:
Eventually what will make your cake hard frozen is large ice crystals. For this you should experiment with the final water amount in your cake to minimize it. Resting the batter/dough overnight can help you hydrate the flour easier.

Increase the amount of ingredients that interfere with large crystal formation. To name a few:

Eggs (especially egg yolk) have various properties for  inhibiting  crystal formation.
Instead of sucrose (table sugar) use invert sugar syrup, which is pretty easy to make, that can help slow down the ice crystals forming.
Locust Bean Gum: this is a thickener known to reduce the ice crystals in ice-cream, and also has baking applications. This can be utilized as well.