Having a separate billing address for the services and for the store

Hey, we got into a bit of a tricky situation and would love to know if there is any solution for us from more experienced Shopifiers.

Basically, we are creating a website for the client. We are the ones paying for Shopify itself, for template, domain, apps etc., so we need the billing address because of invoices (taxes) to be set to our address and our company.

But on the invoices / on the website, we would need our clients company/address, as the orders and such will be used for taxes by our client. So tldr, storefront needs to have a different billing address than the background.

Is there any solution for that? Thank you very much in advance,
Dan from EXPANDO.