Always interesting to see how others do things.
My 2 cents...
First, many of the chemical cleansing products mentioned here are not available here where I live in Central Europe. Personally, I use cleaning products from
Enartis (Sanaton and Detersol). While these are sold "over the counter" here, those are both very caustic chemicals and can cause skin burns and skin damage if used improperly. But removes most "stains" well.
For one unable to get special chemicals, then a very strong acidic (add citric acid) solution of KMeta may be the best option.
Side note: It is a general rule of thumb to not use the same wood press basket for reds and whites. Not so much from color leeching, but due to the possible transfer of malolactic fermentation (MLF) bacteria to the white wine if the red wine had any MLF before pressing. Most white wine should not be allowed to undergo MLF and once wood gets MLF bacteria, it is difficult to remove it and prevent contamination of white wine.
Hope this helps.