Okay… just using what I saw in Black Adam with some other ideas:
It starts simply enough, with the JSA finding out about a plot by Vandal Savage and Felix Faust to open up the Underworld for an army of conquest/Vandal’s favorite wife. There’s some humor within this evil alliance and with Hawkman against them - Felix knows he’s doomed to damnation and a permanent death if he doesn’t carry out his plan, while Vandal is barred entirely from both, and while this makes Felix an absolutely desperate foe easily intimidated by threats of killing him and sending him to hell “early,” Vandal is absolutely fearless and just plain *done* with everything, and has a humorous history with Hawkman.
The reason behind some of that Hawkman’s easy bravado and ease with the idea of dying was because he and Hawkgirl *have* died without finding each other before, and while the love is immeasurable, the pain of one of them watching the other die after a too-brief reunion is excruciating; Carter is fine with the idea of not going through that pain again in this life…
So when he sees Kendra Saunders among a Doctors Without Borders unit deployed to the region and then recognizes as Hawkgirl’s reincarnation her mid-flirty conversation, he has a full blown panic attack, and is doing his best for most of the film to avoid contact with her at all, even as she is uncovering some of her own gear when Savage captures her and frees herself.
Part of the conflict of the film would be that as the two heroes get their bearing and backstory straight, and deal with the interpersonal conflict caused by that, Savage offers to free them from their curse and Faust offers to permanently kill off Hawkwoman’s reincarnation soul before it fully merges with Kendra.
So while the external conflict would be a driving action for the villains, the main attraction would instead be the romance and desperation to escape tragedy for our heroes.