Open Tools/Account Settings, select B in the left pane, then look at Outgoing Server (SMTP) in the right pane. Is B set to send on the smtp server that is defined with A's User Name and password?

Yes, B is set to send on the SMTP server that's defined with A's User Name and password.

You may already have an smtp server defined with B's credentials in Tools/Account Settings, Outgoing Server (SMTP) at the bottom of the left pane. If so, switch B to send through that server. If the B smtp doesn't exist, Add... it in the Outgoing Server (SMTP) section.

The Outgoing Server (SMTP) is currently only showing A's User Name and Password. To confirm your advise: I'll set up a second entry in Outgoing Server (SMTP) using B's User Name and Password. Is this correct? I do appreciate your help.

Yes, if the smtp for B wasn't created when B was added to TB, add it manually and make sure it's selected as B's outgoing server. You will be prompted for B's password the first time you send through the B smtp; check the box in the prompt window to save the password in Tools/Options/Security/Passwords.

Eureka! It worked. Many thanks for your help.