Database Mail stops working after browser service stopped
It has issues because it would use the instance name rather than a port number, and the Browser service is required to do translation from instances to ports. Your other apps must be hardcoded to use the port numbers (or have aliases set in their SQL Client settings, or something similar).

Anyway when a mail is sent this triggers MSSQL\Binn\DatabaseMail.exe to run. I wasn't able to find out how it specifically works out what the default server name is, but you may be able to reconfigure it by creating an MSSQL\Binn\DatabaseMail.exe.config like this on each node, and including a port number:

&lt;configuration&gt;
    &lt;appSettings&gt;
        &lt;add key="DatabaseServerName" value ="LocalServerName" /&gt;
        &lt;add key="DatabaseName" value ="msdb" /&gt;
    &lt;/appSettings&gt;
&lt;/configuration&gt;