If you gave your credit card company an address that you say is your primary address but isn't, you'd be committing fraud. This might never be caught under normal circumstances, but if you ever defaulted on your credit card debt and they were unable to collect, you could be hit with criminal charges - the money you got from them wouldn't be a debt anymore, it would be stolen money obtained through fraud. (The credit card company might consider people without permanent addresses to be high-risk, and not normally extend credit to them.)

Also, the state or country that the address was in might come after you for unpaid personal or corporate taxes.

The service you are looking for is offered by corporate office rental companies.

If you are a nomad currently, you should ask fellow nomads among that community for different providers they know of and what their experience/s have been.

(you'll normally need a business, etc. to rent a virtual office and well... it's quite a process but probably worthwhile if you don't live anywhere for more than a few months)

Maybe this should just be a sub-category on Airbnb?