I started to reply to this at one point, then realized that you're in LA and I am familiar with the laws in VA for this type of thing. I'll still give you my experience and let you draw your own conclusions.
Years ago I was a political appointee to a local county commission. I won't bore you with the details, but at one point I was hit with a FOIA request for my personal email account regarding all commission-related correnspondence between myself and the other eight commission members. I had to turn over copies of all emails unless they were related to 1) personnel issues (none were) or 2) privileged/confidential correspondence between the commission and the county attorney (again, not applicable). So, I turned everything over.
So, my advice as a non-attorney for you would be to look at the content of these emails in the context of the FOIA law in your state. While you as an individual are not subject to it, the mayor certainly is. If the content is releaseable under FOIA then there shouldn't be any issue with you publishing them.
Again, IANA lawyer, and this advice is worth exactly what you paid for it.