Before some posters get too carried away by F/A-XX sizing and engine options, consider that the F414 and F110 have core airflows that are actually not too different. The F414 with a BPR of 0.25 and fan airflow of 170 lb/s has a core airflow of 136 lb/s, while the F110-400 on the F-14B/D with a BPR of 0.87 and fan airflow of 269 lb/s has a core airflow of 144 lb/s, while the current F110-129 is about 153 lb/s. Incidentally, the F414 core airflow is also quite similar to the F100-220 (228 lb/s fan airflow, 0.63 BPR, so ~140 lb/s core airflow).
In other words, the difference between a derivative of the F414 or F110 may not be much different in practice especially if the fan is sized to what the Navy wants in terms of performance. An F414 class engine may be sufficient for something that's modestly larger than the F/A-18E/F, while structural engineering advances can mitigate the weight growth. Or if more thrust is needed, sizing the fan to something like 24,000-25,000 lbf of augmented thrust may be sufficient for the F/A-XX. After all, the Lockheed/Boeing/General Dynamics A/F-X proposal of the 1990s (AFX-653) was reportedly to be powered by a pair of PW7000 engines in the 25,000 lbf thrust range.