Can I design an email with BEE and then send it with MailChimp?

Yes, absolutely, and thousands of BEE users do exactly that. There are many ways to get your email into your MailChimp account, including a free "connector" from BEE Pro. BEE Pro is an email design suite. Basically an upgrade from the free editor that you see at beefree.io. There is a free trial (no credit card required) so you can test it out and see if the MailChimp connector works well for you. For more about using BEE with MailChimp, please see: https://support.beefree.io/hc/en-us/articles/360004495852-Exporting-to-MailChimp