How will this play out ? We have some indications - leaving aside attempts to read White House tea leaves.

Jack Goldsmith's political post (a rare bird, since Jack writes mostly legal) from yesterday outlines some issues. I've had respect for Goldsmith since he duelled it out with David Addington (Wiki; read this 2007 NYT piece on Jack, Conscience of a Conservative - which evinced moral courage on his part once he found the right path).

British Bow Out of Syria Intervention, USG Plunging Ahead (by Jack Goldsmith, August 29, 2013):

In a separate story, the NYT reports that “President Obama is prepared to move ahead with a limited military strike on Syria . . . even with a rejection of such action by Britain’s Parliament, an increasingly restive Congress, and lacking an endorsement from the United Nations Security Council.” And, the NYT might have added, without the support of the American people.
The segment below on the two recent polls suggest something of a shift in the American attitudes. That as it may or may not be, I certainly concur in these snips by Goldsmith:

The NYT says that the President “is basing his case for action both on safeguarding international standards against the use of chemical weapons and on the threat to America’s national interests posed by Syria’s use of those weapons.” These rationales are very weak – especially since the President would be violating international law to “safeguard international standards,” since our closest ally Britain withdrew from the fight, since the U.N. failed to authorize force, and since the Arab League does not support intervention.
...
The President is way out on a limb, by himself. Independent of legality, unilateral military intervention in these circumstances is extraordinarily imprudent, and it is hard to fathom that it is being considered by the man who based his case for the presidency in 2008 on his commitment to domestic and international legality, and on opposition to imprudent wars.

The administration seems to think that the costs of going forward in Syria are small because the planned strike will be “limited.” But even assuming that a limited strike does not produce terrible second- and third-order consequences in the region, it would still be self-defeating because (although it is limited) it would be contrary to international and domestic opinion and (because it is limited) it would bring few benefits in terms of punishing Assad or enhancing Obama’s credibility.
Jack addressed "why not congressional approval":

This is very dangerous territory for the President. Forget the Constitution for a moment. Why won’t the President pay the same respect to American democracy that David Cameron paid to British democracy? (I offered answers to this question a few days ago, but the question is much more poignant now that the British Parliament has spoken against intervention.)
Jack's five answers were (adding numbers to them):

1. has a very broad view of his unilateral war powers;

2. this military action is being rushed, and formal congressional approval is not a priority in light of the President’s self-induced credibility crisis and the overwhelming military and diplomatic demands of planning the intervention;

3. the White House doesn’t want to expend (or doesn’t have) the resources that seeking and winning congressional approval would require;

4. it doesn’t want to suffer through the formal national debate;

5. and it fears it might lose the debate (either outright, or with a limitation on presidential power), which would be politically and legally humiliating.
I'd add that the White House has as much as it wants from the key congressional leadership. Here, for example, is the position of Michigan's senior senator, Levin statement on Syria consultations (Thursday, August 29, 2013)

WASHINGTON -- Sen. Carl Levin, D-Mich., chairman of the Senate Armed Services Committee, issued the following statement today following consultations with the Obama administration on the situation in Syria:

"I have previously called for the United States to work with our friends and allies to increase the military pressure on the Assad regime by providing lethal aid to vetted elements of the Syrian opposition. Tonight, I suggested that we should do so while UN inspectors complete their work and while we seek international support for limited, targeted strikes in response to the Assad regime’s large-scale use of chemical weapons against the Syrian people. I appreciate the administration’s continuing efforts tonight to consult with Congress about the situation in Syria, and its commitment to further consultations with Congress."
Note that "we seek international support" (thus, the effort to bag the UK is probably going to continue); and "consult" and "consultations" do not mean formal approval - even though 80% of Americans want exactly that.

- to be cont. -