My company has a stand-alone solution for registration. Can this be integrated with your LMS?

Yes, but it is expensive to set up. A more cost effective and robust alternative is a shopping site like Shopify that has a back end system for receiving payments and integrate that with the LMS. BHE can set this up, so that all of your client data stays with your company. If you already have an e-commerce site, you can add registrations manually to your LMS slice.