Share shpping cart (WOO) between 3 sites

Hi, I have a client who wants me to set up 3 different WordPress sites. But he wants them all to be linked by 1 database. The main purpose is to support 1 shopping cart for all 3 sites.

I don’t think the multi-site mode will work because he wants 3 totally different URLs, with 3 different styles. So they appear separate but he can pull various product categories to appear on each site.

Can this be done please? And how would you do it?

Thank you, Julian