I think the fact that you move the character around during the battle precludes it from that genre. Auto battles usually have two phases, a shopping or building phase and a battle phase. You prepare for the battle in the former but have no influence on the outcome in the battle phase itself.

I was just reading this article about a mathematical understanding of closed time-like curves.

In essence, the argument is that time travel to the past is possible with a degree of free will, but you would not be allowed to alter the past in such a way as to remove the motivation for traveling back in time. E.g., it would be like Futurama where Fry kills his grandfather, but he impregnates his grandmother, thus allowing himself to be born. The idea is that the timeline would correct itself and ensure that your future self will always return to the past.