Why do dollar signs matter? Is this not a currency outside the usual system? It has absolutely no relationship to dollars except for exchange. It's as closely aligned to Yen (¥) or Philippine Pesos (₱) which both have horizontal bars.

Additionally, the sign you suggest is closer to a Cifrão than a dollar sign as it has two vertical bars. Unfortunately it's common for people to arbitrarily use two vertical strokes on a dollar. If this were to happen with the Bitcoin, then there would be confusion with the Thai Baht (฿).

Instead, use a horizontal stroke and the ambiguity is removed. The symbol doesn't have any close matches in current use or in Wikipedia's 'historical' section.