The City of Charleston will not allow AirBnB rentals to anyone who is not an owner/occupant and even with this the ordinance is very restrictive (ex., 4 adult max). 

Most of the HOA in our area (Jersey shore) have some type of restriction to limit rentals. 

I'm aware that there are restrictions/limitations with this type of setup.

So, I think I'd be restricted to getting a California license.

Depending on what tax bracket you are in, and whether or not you are subject to passive loss restrictions in the eyes of the IRS (i.e. make less than $150k per year in active income, or qualify as a professional real estate professional), that annual total would then be added to your federal tax return.

They have restrictions on what types of properties they take on, but all of those are listed on their website.

Being in the Bay Area, this may be a bit restrictive based on housing prices and available equity.

However, I am concerned about restrictions around using an equity option and then moving out of a primary residence to another.

Are you referring to restrictions on what you can and cannot invest in using your SD IRA? 

In NJ under the Anti-Eviction Act, a landlord cannot impose a rent increase so large that it is "unconscionable", meaning an increase that is extremely harsh or unreasonable.