One approach to achieve the described behavior is by having an additional field in the database, something like "orderPosition" that contains a value which can be used to order the rows.
Then, in the Button click event handler, you can update the database record, by accessing the selected row and swapping its orderPosition value with the other row's located above or below, depending on the direction you want to move it.
Since this is an urgent request, I advise that you open a formal support ticket where we will guarantee 24h/72h response time.
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