Blehhhhhh writing problems…
These are things I’m sure you’ve thought of, but on the off chance you haven’t…
In order to avoid and/or delay the police in your story, could the murder happen in a remote/isolated location? Maybe there’s some sort of weather or situation that causes trouble that would keep authorities away (extreme storms, blizzard, roadways blocked, etc). Or it could happen during a big, (potentially connected?) crisis or event that draws a lot of extra police away from your crime scene.
I’ve read literally one Agatha Christie book, so I don’t know if this is a common trope, but maybe those involved in your complicated murder mystery, at least for a time, agree to not involve the police (I’m thinking like in the movie Clue, that they all have something to hide). That could at least delay the police doing their policing.
Of course, there’s always evidence tampering. Maybe the who in the whodunit works for or with the police and has access to the evidence. Perhaps they have an accomplice to get rid of things for them. The results of any DNA analysis (which… tends to take some time anyway, I think?) could be inconclusive, there could only be partial fingerprints, the victim could have been seen getting into several altercations before their death and have several samples of DNA on them because of it.
Your protagonist could be friendly with the police and offer help them on this case (because it no doubt has some special meaning or significance to them?). They could get in the police’s way time and again. They could temporarily conceal evidence until they figure out what it means, then either hand it over sheepishly or “find” it conveniently or something.
As for what the police will be doing, they should probably be trying to use their police techniques to solve the crime, which would probably be talking to witnesses a lot; digging into the victim’s history to find something that could get them killed; probably looking into the pasts of some of the witnesses, one of whom could be the killer (depending on what sort of setup you use for your murder mystery); finding any relevant surveillance footage (where applicable); that sort of thing.
(And, yes, I realize that I’ve pulled just about every single trope from every single whodunit show and movie I’ve ever seen or heard of, but they’re not overused tropes for nothin’!) (Also, I bet you could take an overused trope and give it a fresh Alice spin to make it awesome.)