For the RitR, is one player allowed to take a "Base Reward" after a pivotal battle? Specifically Demonstration of Force?

We would likely rule that you can't - but you could certainly rule it otherwise.

The argument against is that "Base Rewards" on page 21 of the RitR rules says that "Each campaign round , one player from each team may choose its reward from one of that team's base locations..." The Pivotal battles section (page 24) states that "Pivotal battles occur outside of the normal campaign turn structure..." So, they aren't a "round" for purposes of the Base Rewards. Of course, the problem here is that the rules themselves talk about "turns" not "rounds" except for this definition, so it's a little unclear if the use of "round" is intended to be larger than a "turn" and thus inclusive of the pivotal battle.