I don't think you can. I use the SUM button and then F2 (edit) to change the SUM to AVERAGE. You could write a macro to do that and put it on another button. But what's the point? The SUM button is designed for initial set-up. What is the underlying requirement you are trying to meet?
BTW, if you haven't tried it yet, you can click the sum button twice for the (almost) same effect as clicking it once followed by the enter key. The only difference is the selection stays on the cell with SUM formula and does not auto-advance (assuming auto-advance is active).