carbon is not the only thing that will filter tannins out of the water. Flosses/fibers and some water polishing products will as well.
Black water extract will be filtered out in about a week. It's a constant additive. Meaning you need to stay on top of it weekly at the least.
Peat is a longer lasting approach. I was changing ours out about once a month when Ph levels started to rise.
DW is the best, continuous source. Yes it will eventually stop leeching tannins but that takes a longer period of time.
I've done all 3 of these on our 40g, some might remember the posts about Ph issues. I personally have found that by keeping up on peat in the filter and not using any water polishing media (carbon or other forms) that the tannins leeching from the DW are keeping the Ph in the tank at 7.0, 7.2 is when I change it again. The neons the OP is talking about do prefer, like the rams we kept, a lowered Ph value, hence the tannins. (also they like the darker water) But, as I have also learned, the hard way I might add, keeping the Ph steady is better than mucking with it at all by adding additives. So it's up to the OP to decide use DW and let nature take it's course, or have to keep a constant "flow" of additives on hand.