Can an AJAX response set a cookie? If not, what is my alternative solution? Should I set it with Javascript or something similar?
According to the w3 spec section 4.6.3 for XMLHttpRequest a user agent should honor the Set-Cookie header. So the answer is yes you should be able to.
Quotation:
If the user agent supports HTTP State Management it should persist, discard and send cookies (as received in the Set-Cookie response header, and sent in the Cookie header) as applicable.