It would seem to me that this goes beyond just character evidence. It would corroborate the validity of GZ's suspicions that night of a person who appeared to be casing apartments and thus the NEN call.
I think the evidence should come in, for various reasons. However, there is a counterargument that it's not relevant relative to the attack, and the issue is about the attack, not about a burglary. Nelson is a reasonably skilled "spinner" of her own right, and is capable of misreading law to suit the outcome she wants to obtain. I think all judges have that quality.