I'm working my way through the early Weissinger issues and I have to admit, I didn't realize how odd it gets. Batman exposing his secret identity is just part of it. Both Batman and Superman are characterized horribly in several stories, and some of the wacky stuff that works for Jimmy Olsen or Lois Lane is completely misplaced with Batman as the co-feature.
Consider the topic of mental illness. In WF #143, Batman is accidentally struck by a bullet that ricocheted off Superman. At the hospital, he shows signs of depression:
Superman decides to cheer him up by letting him solve a mystery. So that Batman will believe he's really needed, the mystery is set in Kandor, where Supes doesn't have his powers. A friend of Superman there is in on the gag. But does Batman brilliantly deduce that he's being patronized?
Of course, something's gone wrong and suddenly the menace they're facing is real, but Supes can't convince Bats, who snaps:
Eventually Batman is convinced of the reality of the menace, defeats it, and manages to shrug off his "inferiority complex".
Weissinger appears to have been copying a bit from the Marvel comics, which often featured the heroes bickering. The difference was that in the DC Silver Age universe, everything had to be restored to normalcy at the end of the story, so it was always some sort of temporary mania.
Or alien hypnosis as in WF #145:
Which causes Superman to snap:
In WF #148 Superman decides to check the security of his secret identity. He uses a "selective amnesia ray" to eliminate Batman and Robin's knowledge of his secret identity, so that they can try to crack the mystery. They do this fairly easily, wounding his pride. So he uses the ray on himself to create a challenge to discover Batman's real identity. But the obsession becomes a distraction from his real work:
So Batman decides to let Superman win:
In WF #150, Batman decides to check out Gambler's Isle, a legal offshore casino. He suspects something's crooked because people are losing their life savings. But get how he checks whether the roulette wheel is fixed:
It's not, and so Batman engages in a little "mathematical" gambling:
Hitting on 17? Apparently the great game theorist didn't play a lot of blackjack!
It all turns out to be a plot by gambling-mad aliens to lure Superman and Batman to their world. As all aliens must be, they are a different color:
But (conveniently for plot purposes) their women are not:
Yep, another LL girlfriend for Superman.