Stop Automatic Checking of a Mail Account

I have my own domain. I have a mail account there and forward the mail to iCloud for filtering and storing. My web host cannot automatically delete the mail from the mail account so I end up with 2 versions of all my emails, 1 version in an iMap store which I manually delete and the other version in my iCloud inbox. A "forward" mailbox is no use because I then cannot send emails using my domain's SMTP server.

So my question is, how can I have my Mac Mail NOT check the iMap store in my domain for emails unless I ask it to, i.e. manually. This used to be an option in Mac Mail but no longer is. If I simply take out my credentials I'll get asked for them dozens or hundreds of times a day. And I'd like this to work on iOS as well!

Alternatively is there a mail host that you know that will forward and delete if that is what you want?