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To have it centered in a panel, create a sub-pannel with 3 panes, make the middle one xy, but the outside two variable and set the min/max large enough on the outside panes with the slack distribution define properly and it'll keep the middle pane the same size and distribute the extra size to the outside panes, thus centering your button in the outside pane.
Would relocating left and right pane to the top and bottom respectively achieve vertical centering? If so, how about using four surrounding panes to achieve simultaneous horizontal and vertical centering?
yes, all of that is possible
Not that it really is, and I think what's suggested is a clever way to achieve Timur's goal, but couldn't this be the beginning stage of a wonderful Rube Goldberg's device?