Historically, many of the Democrats who voted for the resolution were not in favor of invasion, but felt support for the President was necessary to get Saddam to back down (and said so at the time). So, the premise is faulty.
If they were weren't in favor of the invasion they should have voted against it. By voting for the invasion they put themselves on record as supporters of the invasion. If they were bluffing they lost.
This is what is called "historical revisionism", friend. It is ascribing positions today that are different to what occurred at the time. Most people, in Congress and in the public, believed Saddam Hussein would back down. The fact that he actually did was ignored by the Bush administration and did not become public until after the invasion occurred. Perhaps you remember that?
When, by the way, was there a vote for invasion? [Hint: there wasn't. H. J. 114
l] There was an authorization for the use of force if necessary
. The decision to invade was Bush's alone. To assert otherwise is revisionism. It is the kind of hyperpartisan factual manipulation that gives partisanship a bad name.