Apple Watch: Ring Close Notifications even on Silent and Do Not Disturb
I'm not 100% sure what you want to do can be done via the Do Not Disturb function, but the place to check is within the Watch app on your iPhone. 

Do Not Disturb


On your iPhone launch the Watch app
Swipe up and tap on General
Tap on Do Not Disturb
Now make sure you enable the Workout Do Not Disturb option
If it's not already, you may also want to try disabling the Mirror iPhone option


If the above doesn't work, then try manually disabling Activity Notifications as required (see below).

Activity Notifications

Before your morning workout:


Launch the Watch app on your iPhone
Tap on Notifications
Tap on Activity
Disable the Goal Completions option
Disable anything else you may want to


I think this should work, but then you'll need to remember to enable it again after your workout. Of course, you could just keep it disabled and use one of the Activity complications to monitor the status of your rings throughout the day.