Never mind. The results are inconsistent. This works sometimes, and sometimes not. Apparently the name gets stuck in memory, but further fiddling with accounts or other random email acts can make it revert without warning. I'll leave the the original post in place as a kind of history of what doesn't really work.
After a great deal more experimenting, I came up with this almost satisfactory work around.
Remember how the GMail account would let you change the name? Well, apparently as long as the GMail account is set up and the mail is turned on, it will override the name in the Exchange Google account. Or, at least it does in mine.
This is what I did. I set up the Gmail account with the name I wanted. Turned off everything except the Mail. Went into Settings > Mail, Contacts, Calendars >> Fetch New Data >> Advanced, and changed the GMail account to manual.
Now, as long as I don't open the GMail inbox, or GMail account it never fetches the mail. This prevents duplicates. After half a dozen experimental emails the All Inbox did not force a manual check on the GMail account.
Of course if you do select the GMail Inbox or account, you're going to end up with some duplicates. When I did this I got one listing for the email in the All Inbox with a number beside it. When I tapped on it I got both emails listed. This is why I said it was an almost satisfactory work around. You can make more work for yourself if you get careless. If you don't normally stray outside the All Inbox or the Google Exchange account, you should be relatively safe doing things this way.
As a further experiment I went to the Default Account setting under Settings >> Mail, Contacts, Calendars and tried sending from the All Inbox with both the GMail account and the Exchange account. As far as I know this setting only affects the sent mail address if you are in the All Inbox.
Both methods worked, however the email seemed to go out faster using the GMail account to send. This did not affect where or how the email appeared when it was received, just how long it took. I'd say the GMail was a few seconds at most. The Exchange account took at least two minutes.
Correction: Just noticed that using the Google Exchange account as default did not send the email with the name. Have no idea why, since when I'm in the Google Exchange inbox it worked just fine.
Last Notes: My Google Exchange account does not show a name at all. It shows my google email address in the list. I can only get a name using the workaround just described. To me that means that there is still something different in how we have things set up. No idea what. I use a "
[email protected]" address. Other people with older GMail accounts have "
[email protected]". Maybe you do? This may work differently for all I know.
I discovered this anomaly while trying to set up my GMail using the Other Account settings. I started to write this email based on that, but decided to experiment with the GMail account first, because it's a lot easier to set up. I saw no difference in behavior.