If you're buying a variety of groceries you should be able to arrange them in your bike bag to provide considerable insulation - this can either be instead of a cooler or in addition to putting things in one.
Also consider that some frozen items can be allowed to thaw without harming them while others (e.g. ice cream) can't.

So on the bottom and sides of the bag put items that won't crush easily but do insulate (cereal boxes, canned goods, pasta, etc.), next refrigerated items like meat, then place some of the frozen things that can thaw (frozen vegetables, juices, etc.) around things like ice cream that can't. Finally place good insulators that are crushable on top (eggs over the frozen things and then loaves of bread over those). So the things that need to stay frozen are well-insulated by being in the center of the bag and surrounded by other cold food. In turn, the things that should be refrigerated are kept cold by being near frozen items. This should easily handle a trip of 45 minutes to an hour with a full grocery load.