Point is....a good discussion to hear different views.
My view again for the OP is not to do core specific exercises. But if he has pain or discomfort on his bike...focus on matching his bike fit to level of conditioning...in his case lack of conditioning and start riding more. He will develop strength through riding. Once he starts riding faster and gaining fitness with a less aggressive bike set up, he can likely tolerate a more performance oriented fit...less setback...a bit more reach and drop which natively puts more weight on the hands.