Done my google search. No mordant for beetroot. Cochineal is a colour that can be set with a mordant. Spent my xmas break tie-dyeing. The die I was using comes in a range of colours with the Alum mordant mixed in with the dye powder. Add cold water and use within 20 minutes. Let set for 24 hours in warmth and wash off excess dye. Might give them a go. They attach chemically to the cellulose molecules in natural fibres so should work well with wood. I will do the mad scientist thing tomorrow and report back.