When Batman leaves the spear the first time, it's under the understanding that this area is his domain. Literally, in the sense that Wayne owns the derelict building, but practically too because he had to be confident there would be no innocent collateral damage to his war with Superman and he was bringing heavy weapons to bear. He would have made sure there were no squatters, trespassers, etc. and he must have been doubly confident in that belief to BRING Doomsday to the abandoned port. So the FIRST time he leaves the spear, it's like leaving your hedge clippers on your porch or lawn... no, it's not where it's supposed to be, but you're confident it'll keep until you get around to it.
Lois, not knowing Batman's preparations to the same degree, takes extra caution to conceal the spear until it can be more permanently dealt with. A little like your out-of-town aunt not knowing the neighborhood and thinking it necessary to try and conceal your clippers in an empty clay pot.
However, AFTER Doomsday, there's no question that the authorities are going to be combing over the location and considering the effects of the fight. At that point Batman's lost dominion over the arena so he wouldn't leave the spear laying around. This time, he brings it with him, along with the body of Superman, but- likely- he leaves the body of Doomsday, lest he start a custody war with the authorities over whether that's something he can take. Given that the authorities are likely to arrive any moment, he probably offers to provide a ride to Lois and Diana.
Lois accepts the offer, which is how Bruce works out the logistics of getting Clark's body to Smallville and paying for the funeral anonymously. He can also arrive at this information by stalking the Martha he saved and who was taken in by police. If GCPD can't trace her identity (which is likely considering she's an out-of-state rural law-abider without a rap sheet) and she doesn't give it, they have to release her and she can return to Smallville without a substantive record of police contact. However, this is creepy and it makes more sense for Lois to go with Batman and explain their situation than for her to stay at ground zero, alone, crying, and inexplicable to the arriving authorities. Lois explaining during the ride also explains how Diana shows up to pay her respects. Otherwise it's too remote to show up even on Batman's invite.
Now that Batman has the spear, the question is: What happens to it?
Consider what Alfred first proposes:
To keep it out of Luthor's hands. To destroy it.
At the time, Superman was perceived as a hero and Alfred says that no contingency against Superman was necessary. This sentiment is shared again and again throughout these films, Senator Barrows:
Last I looked, the only one of those flying around up here was Superman.
You can call me whatever you like. Take a weapon of assassination and call it deterrence. You won't fool a fly or me.
We lose a national hero...
We got lucky with Superman. He shared our values.
Under that lens and at that time, it made sense to destroy the Kryptonite so that no one could use it. However, after the advent of Doomsday and the utility the spear provided in that fight, it's hard to believe Batman would just destroy it. Especially since he believes another alien threat is coming.
Therefore, most likely, Batman has the spear somewhere secure and secret. Even if the cave is where he'd keep it in tradition, because his identity is exposed and out there to the likes of Amanda Waller... and because the stakes are so high... the spear is probably kept in a custom location and not the cave.