Will you be refusing to answer the Census? This is probably much less intrusive than Census questions.
I really didn't want to get into this because many seem to be so emotional about this but, I am more or less an insufferable libertarian butt crevasse, so I get the serious privacy concerns here, but it's not even clear that the U.S. Constitution actually protects privacy. It certainly doesn't explicitly protect it. Although it is certainly possible that it's a penumbra right as I know you know. But, I'm not really sure what rights this violates and even if it does violate one, you know that it's a balancing test. And you could probably make a good argument that this is narrowly tailored to achieve a specific goal during a state of emergency. It's not like NOPD is going to be at the door checking your papers. It's likely that the information you provide won't even be verified by anyone so you could just give a false name and number if you find it that big of a deal.
I was a lot more bothered by LaToya telling NOPD to set up pretext check points to check for seat belt violations and then having NOPD get in people's faces to remind them to stay home. I mean, that actually implicated the 4th Amendment since it probably counted as a search done by NOPD. You know they were looking for any excuse to search any car that they though met the profile of a drug user/dealer. On the other hand, this isn't a search and it isn't being done by NOPD or any government official. So, I don't think the 4th Amendment would apply.
And I'll add that it's no different from what Florida is doing at their border and nobody seemed to express outrage when buzd explained the process he went through when entering Florida.
Edit: Also, you would know more about this then me since I think you are a former Public Defender or DA, but couldn't NOPD just subpoena records from stores and then get information from credit card receipts, information provided for bonus reward programs, etc. if they wanted to? I mean, it would be much slower and likely not effective in contact tracing because it is so slow, but I think they could do it anyway. And, I get that at least you have the procedural protection of having a judge sign the Warrant, but this way is much faster and neither NOPD, the City, or anyone else, other than store employees, may ever see the information.