Turns out the Poor Man’s Veneer Press I made earlier in the week works perfectly. The chessboard came out almost perfectly flat. Used cold press veneer glue this time and no worries about getting the veneer tape wet to remove. A bit of work with the card scraper and off it came.
How to sand it smooth: Scribble all over the surface with a pencil, making sure to have marks crossing all edges. And sand until all the marks are gone. No worries about sand-through, thanks to new press. Me a happy wood worker.
Also, Chechen and Holly for woods on this board.