IMO the US Constitution is unlikely to exert a meaningful influence on the outcome of Bitcoin. Especially if the DOJ, FBI, NSA, CIA or any other US Federal TLA's are involved -- US Federal LEO's mostly think the Constitution is an historical relic and anyone taking it seriously is probably some kind of "Taxi Driver" type whose phone should be tapped as a safety precaution.
If I had to bet, I'd bet that a US Federal campaign to destroy Bitcoin will happen sooner or later. Because Bitcoin offers a kind of anonymity it will be labeled as "money laundering." However I suspect the Feds will go to great lengths to ensure that Bitcoin never gets its day in court, unless, of course, an opportunity arises to prosecute an individual or group who is jurisprudentially clearly guilty of a bunch of crap unrelated or tangentially related to Bitcoin (i.e.: wire fraud + racketeering + tax evasion + money laundering, all using Bitcoin).
As others have posted, it may take them a while to figure out what Bitcoin actually is. But don't expect Feds to live up to their infamy as misinformed dolts in 2011. In the Internet era, facts are cheap. Anyone with access to Google, git, and an abacus has the requisite tools to figure out what Bitcoin is and the Feds will make sure they have at least a cursory understanding before they send the goon squads to round up their citizens.
Here's my reasoning: since 9/11, it has become -- I suspect -- sacrosanct in the Federal LE community that highly traceable, highly seizable money is going to be the cornerstone of a new anti-terrorism and internal revenue enforcement regime, and that protecting the control and monitoring of the movement of all practical money instruments is a a top priority -- much higher priority than, say, enforcing actual laws.
So once Bitcoin reaches a certain critical mass -- and I fear it looms dangerously close already -- its going to light up pretty big on their radar. Even if the real money supply of Bitcoins were to remain unexpectedly small, a high velocity of money could still threaten the control of the Fed and the Feds over their money-power security blanket -- by which I don't mean an actual high velocity of Bitcoin, just the perception that it is
possible for Bitcoin to accelerate in the future without real expansion of the monetary base.
Personally, I'm just grateful for Bin Laden -- that could very well have bought the US an extra six months or more. I wish I were kidding
On the other hand, I think any US Federal attempt to shutdown and outlaw Bitcoin will most likely fail. I suspect that, due to the design of Bitcoin, they will be forced to trample on some fairly fundamental human rights or to implement some sort of Great Firewall to fight any "war on Bitcoin;" ultimately it will be a PR disaster. Eventually some US administration will be smart enough to know when to back down. I could see it being put into some sort of legal-gray-area status or even recognized as legal but with some onerous tax accountancy burden attached to it to make any real-world, practical usage of Bitcoin a
de jure crime even if it's technically allowed.
Of course this is all wild speculation and even if I'm right, the whole process could take decades. But if I had to wager on an outcome it would be something along those lines.