I'm pretty sure it's already been heavily implied that Claude does indeed remember some of the previous lifetime and, more importantly, that he is the one that turned back time. During the last two chapters of the first season he displayed obvious demonic power, back when Ivona was stuck to the pillar. It was this power that allowed him to separate her from the pillar when his holy sword did nothing. Right before he awakened to this power and saved Ivona we got some vague flashbacks of what happened after Ivona died in the previous lifetime, as well as some heavy hints that the person he was unable to save, the one who was "Dead, in the end..." and the one he "wanted to protect the most," was Ivona. The flashbacks we saw, and that this current Claude saw(since he was regaining those memories at that exact moment), seem to suggest that he cared for Ivona greatly in the previous timeline despite her situation, and that in order to get her back he dipped into a dark power that was supposedly chained up. My guess is that it was the Demon Lord's power that was chained, and Claude sought out this power after the Demo Lord was defeated so Claude could turn back time in the hopes of saving her. As for why the Demon Lord would allow his power to be used like this(assuming he had a say in the matter and isn't the one that suggested it to Claude in the first place), this benefits the Demon Lord since it'll allow him a second chance to take over the world or whatever his plan is. Even better, the "Savior" will become corrupted by his power and even become a potential pawn once he's resurrected again. The Demon Lord simply wasn't anticipating Ivona remembering everything and thus her efforts to prepare for his resurrection long before his revival. If she had gone the same route as she did before then the Demon Lord could have just waited till Ivona died again and then take over the grief-stricken Claude's body before he is defeated this time around, maybe by prolonging the war and not directly fighting Claude or Ivona until she reached her limit.