Is there any way to use sleep mode on WatchOS 7 without do not disturb activating on my iPhone?

You can turn your watch screen off and the watches dnd on with “Theater Mode”, it provides much of what you want. It does not initiate the “time to fall asleep” counter, but other than that works fairly well.

As for sleep mode and still getting phone notifications, are they something that you could set up to trigger a text being sent to you? Because if you can, you can set the originating number up as a favorite, and to break thru DnD, and have them come in that way. Sleep mode has the same rules for Breaking thru as normal DnD.

Lastly I would encourage you to really look at what notifications you feel you need to be getting and If you genuinely need those interrupting your rest? I understand some things are needed, I have some emergency things I have to be available for. But I also get how we are getting so used to always be available, we loose sight of how important it is to put it all aside and get real restorative rest and calm. There is considerable studies showing how difficult it is to calm our brains when our phones are nearby us.

Good luck and i wish you good sleep.