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If you seal the cut edges with something gnarly like PU varnish they should be fine. Just make sure there are no areas where water can collect and pool. If it all dries out after use it should be cool.
Get 18mm birch ply with one side faced in melamine / formica (they're just brand names as far as I know) and fake the 36mm edge by gluing and screwing (from the bottom) a baton of the same material along the edges.
If you're buying sheets and doing it all yourself Richard Russell is good, as is Neil Gotz - 07944 618881. He'll source, collect and deliver from the big guys like Laytham that don't want to do single sheets.
If you're getting them CNCed to size, Cut and Construct are the guys to go for.
Question for @dbr or any other makers on here. Is formica-faced birch ply suitable for an undermount sink? Instinct says no - risk of delamination over time - but I have seen it done. Asking for trouble?
If we do go with this as a work surface, who should we buy from? And is there a difference between formica and melamine? Ta!