a work around to send using an HME email you need to initiate - if you have a prior email from them, using your HME:
open that email
>reply or >forward (if you need the attachment from their email)
you will use this to create your blank slate to send a new email to the merchant etc
Delete the subject, and prior content in the email body (I used command A to ensure I get everything in the body such as formatting etc)
now "send" using your HME.
Clunky but reliable until Apple tweaks HME a little more. They are getting there. Amazon has been my beta useing HME since they have a website forum as a back up that logs all customer and merchant interactions.
HME visibility now on reply emails: I've noticed with the last update to 15.2, the HME is actually visible - I just updated my iphone 13 Max Pro and see it wonderfully there now. My latest MacBook 16" M1Max is still on Monterrey 12.0.1, showing I have one update pending for 12.1 and the HME is not visible in 12.0.1 - I suspect 12,1 will also update the HME visibility to match the latest iphone update so they are all visible in Sent folder emails (I'm sure I wasn't the only one to senbd feedback at the value of having this).
You'll see the "Hide My Email" in your receipt email from the Merchant.
When you use reply, it will now show the entire email (to you) so you not only know it's HME but you know WHICH.
If you any reason you reply to an email you've deactivated you'll get an error saying HME is not avaiable. You'd to restore the email to active to resume using it as an HME.
EXCEPTION: If you have a threaded email with the merchant, be sure to use the right thread if it pertains to a new subject. This would be the only downside to this work around.
If you need to start a fresh subject with the merchant, find their online "contact us" form to initiate a new thread for your new subject, and use your existing assigned HME.
There's so many scenarios I can see why Apple delved into this a little slow so they could use the feedback to make the right adjustments on a broad need.
Examples
The email from Amazon I used before I cleared the subject and body (scroll down to a second to see after hitting reply result):

The "reply" email result, after removing subject and body:
[Image Edited by Moderator to Remove Personal Information]