The OP is in so much discomfort he/she is literally unable to ride more than 14 miles in one day. A weak core isn't sufficient to explain this. In general, people on these boards WAY overrate the importance of core strength - most of the chatter about it from most posters on this board is basically cycling cargo culting. They've heard other people talking about it, but don't actually know what it is useful for and what problems are caused by lack of it. Riding a road bike 15 miles without enormous discomfort doesn't require tummy crunches or planks. Enough of these silly, super-confident know-nothing diagnoses. "It's probably your core." Actually, it almost certainly isn't! And if core strength is a factor, which it could be, it's certainly not that alone.

As for fit, the particular case I described may well not be the issue. As others have pointed out, there are simply too many variables. Maybe the OP has a bad back. But fit, as a general issue, is definitely the first place I'd look, not to inadequate core strength.