The Slaughter rule made more sense before the Senate Parliamentarian ruled that Obama had to sign the Senate bill before reconciliation can take place. If that weren't required, then the House could pass their fixes, deem the Senate bill "passed" by implication, and then the Senate could pass those fixes through reconciliation before the bill is presented to Obama for his signature. (If the Senate altered the fixes, the House would have to vote again in order to satisfy the Constitution's bicameralism requirement, but the filibuster in the Senate would still be avoided.)
This would have alleviated the concerns of House Democrats who don't like the Senate bill and don't trust the Senate to follow through with reconciliation after Obama has already signed the Senate bill into law. The Senate would be under no obligation to reconcile the bill under those circumstances.
But since Obama has to sign the bill before the Senate can proceed with reconciliation, the only possible benefit would be to give the House Democrats a political fig leaf. Not really worth it unless a sufficient number of House Democrats are unwilling to vote for the Senate bill but willing to go the Slaughter route. I don't see why that would be the case, since no one would buy that excuse, as Brainster pointed out.
Is this Constitutional? Probably not, I think. But that's far from certain. The House gets to make its own rules, but only within Constitutional limits. Article I section 7 of the Constitution requires that an identical bill pass both houses of Congress. Whether "deeming" counts as passing the bill is unclear. The House would essentially be passing two pieces of legislation with one vote, in order to avoid accountability for voting for the Senate bill. Arguably within the letter of the Constitution, but it sure doesn't seem to be within the spirit of the Constitutional requirement, which is intended, at least in part, to promote accountability of legislators.
Of course, this says nothing of whether it would be politically wise to try this. I think it would be a huge mistake.