Columnaris in its most virulent forms kills in a few days or a few hours. When did the wound go white? If it's been white for a while, there is still hope, especially since the mouth and fins are not involved. If it's progressing that slowly it might be worthwhile trying to treat it. If it's just gone white, it was probably something else that has stressed the fish and caused infection with columnaris, in which case the columnaris will be rapidly fatal.
If you do attempt to treat but there's no marked improvement in a week, if you can't get the meds or you think the fish is really suffering then I would agree with what most people have said, it's best to put him down.
If you have to do it, take him out of the water and cut down directly behind the head with a sharp knife. This kills instantly. If you can't do that, the next best thing is clove oil. Get a cup of water, add about 15 drops of clove oil and shake it well. Drop the betta into it. It also kills virtually instantly. If you can't get clove oil, you can also use very cold water. Put a cup of water in the freezer until ice forms on the surface. Break the ice and drop the fish in. I personally hate this method and would not use it on a betta, if I had no clove oil I would decapitate. Ice is only humane where it kills within a few seconds in my opinion and I think a betta is probably too large for it to be a nice option.
None of these are perfect, veterinary anaesthetics are the only 100% humane way to euthanise a fish. But they are not available to many hobbyists and there's a lot of debate about how it should be done. Personally I would recommend clove oil as it is instantaneous if used correctly and I know a lot of people cannot bear the thought of cutting their fish's head off. I know I would really not like to do it.
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