Regarding the one failure:

In general, if you're connected via one ISP then you have to send your mail through that ISP's mailserver.

Usually, the only exception is where you have an account on another ISP that lets you authenticate yourself on their mailserver in order to send mail.

Authentication can be done in any number of ways - SMTP username/password is common - but essentially you want to see if any of your ISPs support 'Authenticated SMTP', and if so you'll be able to (with a reconfig of your mail client) send all your mail through that ISP's mailserver, regardless of who you happen to be connected through at the time.
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simon