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Yea they would, regardless of the spending bill that was passed the house and senate have to approve raising the debt ceiling and allow spending beyond it.
The debt ceiling is a completely separate issue
Fourteenth Amendment, Section 4:
The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned. But neither the United States nor any State shall assume or pay any debt or obligation incurred in aid of insurrection or rebellion against the United States, or any claim for the loss or emancipation of any slave; but all such debts, obligations and claims shall be held illegal and void.

The debt limit is unconstitutional. Congress already authorized the debt when they passed last years' budget. The 14th amendment declares that the validity of that debt shall not be questioned, and any legislation preventing payment of that debt is unconstitutional. 
The Treasury department should just declare that the 14th amendment renders the debt ceiling unconstitutional and they will continue to make interest payments on the federal government's debts. >


The Treasury department should just declare that the 14th amendment renders the debt ceiling unconstitutional and they will continue to make interest payments on the federal government's debts.

Thr treasury doesn't have this power, and for good reason. Any entity should never be the writer/interpreter for their own scope of power
Edit: to clarify for people responding to me

The spending bill is not "law", it is a concurrent resolution and does not have "force of law" 
The 14th amendment does not inherently grant any powers to the treasury, and any interpretation that the president could use it to raise the debt ceiling is just that - an interpretation. Until there is historic, legal precedence, it is a moot point, especially considering our current justices.

Edit 2: also, I glossed over this just because it's so "wrong" - the treasury can't "declare" anything unconstitutional. They can sue another department to comply with certain laws, but there's 1. No entity/department to sue, unless you think they can sue congress and 2. There's no laws compelling anyone to raise the debt ceiling
The Treasury can argue that the 14th amendment grants the power to raise the debt ceiling all they want, even if it was the case it would have to be passed by the house, the treasury has no say in it at all >


Any entity should never be the writer/interpreter for their own scope of power

They wouldn't be. Congress already voted on the spending measures and authorized the Treasury to pay them.