"doubtful argument" is correct and usable in written English.
You can use it when you want to express doubt as to the validity of an idea, opinion, etc. For example: After hearing her questionable argument, I was left unconvinced and with a doubtful argument.
But it seems doubtful such an argument – that the deal, while imperfect, is the least bad option on the table – would be likely to persuade Republican critics.
Aside from the fact that, as an American, I don't find that very appealing, I'm very doubtful of that argument for a few reasons: (1) Chinese manufacturing will automate and may do so much more rapidly than was the case in the US because they simply have to import the technology, not invent it.
It is somewhat doubtful that this argument can be successful.
It is doubtful whether such arguments, discourses, or reasonings are purely epistemic (though many of their premises will be epistemic).
(It is, however, doubtful that these arguments were historically ever intended as freestanding proofs that would be convincing to atheists. For an illuminating discussion of the functions of such arguments in pre-modern and early-modern contexts, see Clayton, Parts II and III).
Among partisans debating the Bergdahl deal from right and left, it is doubtful that the cacophony of argument changed a single mind.
On the first argument it is doubtful whether the big Asian economies really have enough fuel in the tank to pull the rest of the world out of a slump.
Justice David Souter, writing for the court, rejected that argument, finding it doubtful that Congress intended fiduciary standards to cover decisions involving treatment made by doctors on behalf of a health plan.
But Justice Margaret A. Chan of State Supreme Court in Manhattan appeared doubtful about the city's arguments, which centered on whether the bicycle tour is a charitable event and the fee paid by its 32,000 participants.
