I just figured out how to solve my own similar problem of outgoing email rejection, this was with Outlook desktop with a non-Cox domain email.
I'm not sure if it's exactly the same combination you're dealing with as I'm not familiar with Thunderbird, but may be useful to others as well.
First, the entry level Cox tech support will have no idea how to deal with this unless you are using precisely a cox.net email. I got the "call Microsoft" nonsense and asked for a supervisor who transferred me to "Tier 2" support, which got me much closer but not all the way.
My situation is I own a domain for a website and email, and need to point to their incoming POP3 server to receive mail, but outgoing needs to route through Cox SMTP at my house. The recent Cox changes now require extra outgoing authentication protocols. So . . .
#1) Mail Servers (I'm using POP3)
Incoming is what my domain hosting gives me
Outgoing is smtp.cox.net (no more "west" / "east" in the name)
My login info is unaffected which is my email@domain.com sort of thing
#2) More Settings - "Outgoing Server" Tab (in outlook)
Check box for: "My outgoing server requires authentication"
I can NOT "use the same settings as incoming", and instead use the "log on using" button, then fill in my Cox.net username and password - I went ahead and created a cox.net email account I didn't previously have associated with my account (not sure if this was necessary but it worked; It defaults to the same username@cox.net and same password. Don't use the "@cox.net" however in this field)
#3) "Advanced" tab
This was the part I figured out myself:
I need to keep the incoming port as 110 or mine won't work anymore; and I do NOT check the box for "this server requires an encrypted connection" which breaks my incoming protocol, (as well as changes outgoing settings).
However, you can still set outgoing authentication that Cox now requires by manually changing the SMTP port to 465, and in the case of Outlook, use the dropdown box to change that part only to SSL.
Jim in AZ