I figured out how to fix this - the real solution
The problem isn't with iOS at all, it's a problem Gmail (a question of implementation rather than a true problem). Specifically, where Gmail collects the information that's used as the display name for Exchange.
Let me explain:
Ideally, a user's Exchange display name in Gmail would mirror the "Send mail as" field that's located in Gmail's account settings just as it does with IMAP. However, this is not the case. That's because the display name used by Gmail with Exchange on iOS is populated from a completely different settings data in Google's services. Instead of using the name that's located in the "Send mail as" field in Gmail's settings, Exchange uses the name data that's stored in the user's personal info profile on the "Google Accounts" page.
Here's the fix:
Access the "Google Accounts" page by logging into Gmail.com, then clicking your email address that's located near the top right corner. A drop down menu will appear, then select "Account Settings" and the Google Accounts page will load.
On the Google Accounts page, click the link "edit your personal info without creating a public profile", then the personal info page will load.
You should see the First and Last name fields. The data contained in these two fields are what Exchange uses to populate users' display name when used with Gmail on iOS. After editing and saving the two fields, an immediate update on the Gmail Exchange server will occur; thus, fixing the incongruent name issue when sending email using your iOS device setup to use Gmail with Exchange.
Note this: Gmail accounts using IMAP on iOS or IMAP on any platform may reflect whatever the user has stored in the "Send mail as" field in his or her Gmail settings. In contrast, Gmail with Exchange doesn't obtain its data from here. Instead, the data it uses is from the information located within the Google Accounts settings page. This explains why some users are able to get the desired display name by enabling their IMAP Gmail and using it to send mail instead of using exchange.
I hope this helps!! Let me know!