Sharing same Inventory on 2 sites. Best option?

We want to have two separate websites – Site1.com & Site2.com – with different pages & different themes but accessing the SAME inventory of products so the admin only has to add/edit/delete the products in one place.

The visitor sees no relationship between the two site. They are mutually exclusive. Only the admin knows the product data is same.

These are flooring products with many manufacturers, makes/models, styles, colors, sizes, installation types etc. It does not need a shopping cart but the would be a nice option for later possibly.

In an ideal world we would also have a flag to select yes/no for viewing on either site such as
Display This Product/Item On:
– Site1.COM (yes/no)
– Site2.COM (yes/no)

This is a move from an old system and is from scratch on new WP or WP Multisite.

Any ideas? Thanks for your help!!