After looking into this more, it looks like there isn't really anything that can be done in SmarterMail.
The information that Matt found regarding <ExternalPhotosUrl> in the autodiscover documentation appears to be unused. Testing against both Exchange 2019 and Outlook.com accounts indicates that <ExternalPhotosUrl> is not being used. Attempts to use the recommended URL specified in the documentation about <ExternalPhotosUrl> result in 404 - Not Found response from both Exchange and Outlook.com.
We also did some extensive testing with the Outlook Mobile app, and it appears that it has some unusual built-in behavior. When testing, we discovered that Outlook Mobile only shows a user profile pic if it is connected to only one account, and that account is on Outlook.com. After adding another account from any email provider, Outlook Mobile began displaying the Outlook.com account with the Outlook icon. Further, when connecting to an account using EAS, Outlook Mobile appears to automatically assume the account is on an Exchange server, and displays it with the Exchange server icon. If Outlook Mobile is connected to multiple accounts from the same hosting provider, or from an unrecognized provider, Outlook Mobile displayed each account with a colored circle containing the first letter of the email address.
With that in mind, it seems that custom images - such as the college logo in the example from echoDreamz - are handled through some unknown mechanism. Given the way that Outlook Mobile functions, we don't really have a way to identify how it is determining that it should use that specific icon.