Having searched quite a bit for this, I’ve found it is possible to use gmail (or any other account) as your “send as” address but retain the push email functionality.
The setup process is:
1. Set up an iCloud account
2. Forward all mail from your alternative domain (gmail, hotmail, domain.com, etc) to your iCloud account
3. On your iOS device, set up a new mail account settings>mail, contacts, calendars>add account>other and use the iCloud imap server and the alternate email smtp server.
Name: Your name
Email: Send as email
Description: Your choice
Incoming mail server
Host name: p99-imap.mail.me.com
Username: iCloud username
Password: iCloud Password
Outgoing mail server
SMTP: details of your SMTP server
Eg, for gmail:
Host name: smtp.gmail.com
Username: gmail username
Password: gmail password
Use SSL: on
That way you can still receive push email to the default iOS mail client while sending as the account of your choice.
The main trick to remember is setting it up under “other” accounts. That way you can use both iCloud for push and a separate server to send as.