Is the one-more discrete log problem hard in the Generic Group Model?
Yes, this was shown in a recent work of Coretti et al [CDG].  Loosely speaking, the lower bound states that an adversary that makes at most $T$ queries to the GGM oracle succeeds with probability at most $T^2/N$, where $N$ is the size of the group. They in fact consider a stronger model where the adversary is allowed arbitrary pre-computation in the GGM. See §5 for more precise statements and Table 2 for a summary of their results.
[CDG]: Coretti, Dodis and Guo, Non-Uniform Bounds in the Random-Permutation, Ideal-Cipher, and Generic-Group Models, Crypto'18