Does the client you're currently using definitely have private key permissions for DM's?
Discussion
*Permanent permission. If you're allowing each time for a DM it will not retain those DM's.
I don;t see anything in the app to specify that
Yeah I was just looking at noscl and I'm not even sure how you would check. When you use nos2x it has a popup that has a few options including allow one time and allow always. Once set I can then see in the settings the following:

I sent you a DM from the bot account. Let me know if you see it and when it disappears
I see it, do you want me to respond?
hmm not sure if that is needed. I see the message I sent myself in Damus and Iris and you see it so now just wait I guess
It’s gone now on Damus . How about you
I still see it in Iris but if I look in other clients where I hadn't previously pulled the message it's not there