You can't create cookies that outlast a private browsing mode session, so if you need to keep cookies from specific domains then you can't use private browsing mode to visit these websites.

PB mode means that no session data is saved to the profile folder, but is kept in memory. Firefox uses two independent cookie jars, one for normal mode and one for Private Browsing mode that is shared among all PB mode tabs and windows. During a PB mode session or in a PB mode window all cookies are kept in memory in a separate cookie jar just like other session data is kept in memory. Cookies saved in normal mode can't be accessed as they are stored in a different cookie jar.