I just stayed at the GH a few weeks ago. Many yatai are across the street, the street food stalls that setup in the evening. Also a short walk from Kawabata Shopping Arcade. Hakata station not far away either, lots of food options there.
Check out White Glass coffee and Full Full Hakata.
The Ritz has Shin Shin ramen and Yatai Mamichan nearby but we walked there from near the GH last fall but didn't have a kid who might not be able to walk so far.