Have you tried putting him back in his own cot after tha 3-5 am feed? To me it seems its more of a comfort thing than he is hungry as the breast is there he will suckle away on it.

My DD will still have 2 good feeds over night, one bout 11:00ish and another bout 5ish. If you were wanting to stop the snacking maybe see how he resettles after that feed using other techniques such as patting or rocking til hes drowsy the putting him down.