The prosecutor would want it in, because it makes Nick look like a disgusting subhuman. Of course, they wouldn't say that, because that's not a legitimate reason for admitting it.
There are a lot of potential justifications for admitting it. Mental state is one. There's also an argument the coprophilia is a signature of his crime. Or that it amounts to a confession of sorts.
The reason not to admit it is basically that it's horrifyingly prejudicial. Anyone seeing it is going to be absolutely disgusted by Nick and basically, the trier of fact (the jury or the judge) will improperly use it to conclude that Nick probably committed the crime because he's just a disgusting person who would do something like that. I.e., he's a bad person and therefore did this bad thing. That's improper fact finding, and evidence likely to lead to such prejudice is inadmissible when the prejudice outweighs the probative value of the evidence, "probative" meaning evidence that makes it more or less likely that material facts relevant to guilt are true.
Basically, it comes down to the judge's decision whether something is more prejudicial than probative.
I think it's pretty much a judgment call. It is highly probative evidence, and it is also highly prejudicial.
Whatever the judge decides is unlikely to lead to a successful appeal. But if it goes to trial and I'm the prosecutor, I want all that horrible stuff coming in and will try to introduce every bit of it.