This app was mentioned in 4 comments, with an average of 1.50 upvotes
Replying to myself with a recommendation for Add Watermark: https://play.google.com/store/apps/details?id=com.androidvilla.addwatermark
This app ticks almost all the boxes on my list! The file selection dialog is a nightmare, but you can pick your photos with your preferred gallery app and share it to Add Watermark for batch processing.
Takes all of 5 seconds once you get the gist.
I use
https://play.google.com/store/apps/details?id=com.androidvilla.addwatermark
You set it up once by long pressing on the words or hitting the settings button. Choose how it looks, whether words/name or a png logo with transparent background and then save and share.
My work flow is Google Photos -> Lightroom Mobile for crop or smudging, clarifying, decision making-> Snapseed for bits of this and that or fine tuning -> Add Watermark. Using share between them is fast but doesn't save as you go so I do a quick save to device then share to next app. Once I pick which images I like, I load them into lightroom mobile, then off I go. I can snap, edit, mark and share in about a minute or two per shot.
Every image on my site is done the same.
The paid version of Add Watermark on the Google Play Store will let you do both functions (free version only lets you do text watermarks).
https://play.google.com/store/apps/details?id=com.androidvilla.addwatermark
I haven't found one for free, but I found this one and it costs only 1,79€ https://play.google.com/store/apps/details?id=com.androidvilla.addwatermark