I was assuming the bat was for resisting arrest by either LE or anyone else who spotted him and wanted to hold him till LE got there.
I'm wondering if he was totally without a phone (having left the burner phone at his sister's house) and therefore not sure what alerts or LE action had yet been taken. He must have assumed the crime would be discovered very quickly. But it would be very strange and paranoia inducing to assume LE were looking for him but not know.
I think we are going to learn the baseball bat was his weapon of choice.
He may have still had it with him as you say to resist arrest or to clean it up at another more secure location. He also could have kept it so he could relive his crimes.
Even if they found blood on Elizabeth's clothing that would not be enough for LE to immediately determine to shift the searches from rescue to recovery.
Blood, and brain matter found on the bat...would immediately let them know she is dead. Even though it will be sent to the lab for forensic testing it does not prevent any experienced LE from recognizing human blood, and brain matter. Imo, they knew what it was even before testing.
As an aside topic:
While some believe this wasnt premeditated, I respectfully disagree. Imo it has all the earmarks requiring premeditation.
Being organized or disorganized has nothing to do with the crime being premeditated. There have always been organized, and disorganized premeditated murderers.
I also dont think he suffers from any mental issues other than he may turnout to be a sociopath. Why that seems to come in prematurely in just about every case still continues to puzzle me when most all murder defendants arent/werent mentally ill at the time they murdered..
Imo, It's very clear he had clear cognitive thinking to not only pull this off without detection, but to cover up his crimes afterwards.
Jmo