An old travel buddy of mine is back in Batumi right now bemoaning the fact that she can't find any cafe serving coffee or breakfast in the city until after 9am.
Are there perhaps some places there that are just a bit harder to find?
I had this problem in Batumi some years ago, except I think we were looking for somewhere around 7am-8am (I don't remember why, I think we needed to take an early mashrutka somewhere).
Batumi is definitely not a "mornings" sort of town. If you want a coffee at midnight though, you're spoiled for choice!
After lots of walking around, we found somewhere that was (if I remember right) opening at 8am but took pity on us and let us in early. I remember it being on the corner of a square / park one block from the beach and beach-facing parks.
After some browsing on Google Maps, I think I've found the place, "Coffeetopia", open 8am-1am. I don't recognise the name but it looks similar to what I remember and the location looks right:
There's also apparently an 8AM-opening ice cream place further from the beach, off an even smaller square, Luca Polare, allegedly open 8am-2am - I'm pretty sure this isn't the one I'm thinking of, as I'd probably remember if I'd had ice cream for breakfast and it's several blocks from the beach, but it might be another option.
While browsing I also found a "Restaurant Ukraine" that allegedly opens at 8:30 AM, that's within 10 minutes walk from both the other options, so this is probably the best area to explore.
Whether these places really do open that early, I can't say for sure... But I believe it is possible. Good luck!