How to remove the anodizing on very small copper wires?

Nothing other than scraping it off with a sharp blade of some sort - but it's not likely to be successful because that's such a tiny wire. Don't be a cheapskate - replace the bloody toothbrush. If it's all that new, it should be under warranty and a replacement should be just a matter of taking it and your sales slip back where you got it.

Oh - you're one of those dummies who tosses sales slips, eh? Sorry, you're S.O.L. Gotta pay for the new one, then...

Anodizing Copper

The fastest way is to burn it, but youre not supposed to. Depending on what size it is, you can tie one end to something and stretch it out and run a box cutter down the side and just pull off the insulation. If you can let sit in the sun and get it warm the insulation will come off easier. If you do decide to burn it, make sure you do it somewhere that you are away from anyone. The fumes are toxic and it smells awful.

Get or borrow a pair of wire strippers that strip small gauge sizes (larger the # smaller the wire) Its probable # 18 or higher. Or you could try and melt it off.

There is a material sold for this purpose called wire stripping compound or enamel stripper. Try Radio Shack or similar electronics supplier.

can you find the begining of the wire and the end and replace it with a small substitute wire (something you COULD solder?)