If one accepts the exclusion argument (no physical event has a non-physical cause), the supervenience argument (no mental event causes any other mental event), and the pairing problem (no physical event can cause a mental event due to the lack of any pairing spatial relation), would one be obligated to accept reductionist physicalism Well, of course we have to at least accept that there is physical-mental causation in order for the argument to work.
(I deleted this answer because its basic interpretation of the quesion was wrong. But most of its content is still germane. So it is back.)
You still have the problem with positions like transcendental pantheism or Deism, who create a third (or even a fifth) substance. For them, the mental may currently supervene upon the material, but only because the material supervenes upon the 'spiritual' or some other determinant with an internal will.
Many of them suppose that while the latter state is permanent, the former may not be. That gives them a secularist eschatology and a notion of heaven, either as some adaptation of Nirvana or as union with an ultimate God. Or it provides them with a theoretical understanding of will that is non-determinist in a universe that is statistically determined. This is not too silly or arcane because folks just asked a question about it https://philosophy.stackexchange.com/a/36062/9166.
They derive support from the basic nondeterminism of matter, which suggests there is a lower-lying determining source. (At least in spirit, for their logic is often kind of nutty, even when they are brilliant, q.v. https://en.wikipedia.org/wiki/The_Emperor%27s_New_Mind where the third substance is 'information'.)
So I still think you need to add the requirement of dualism to get your argument started.
And you still have the problem with alternative notions of causality.
One, coming from positions as wide apart as Hume and the religious notions of 'maya and lila', is that the regularity of physical causation is an epiphenomenon of perception -- not really present, but agreed upon by all those trapped in a shared illusion for their own peace of mind. (Like Folie-a-deux, but Folie à tous). Thus, you may reduce the mental to the physical, only to have the physical reduced back to the mental on a different scale.
Another is the return to the Medieval Christian notion that cause is really primarily teleological, flowing backward from the intended outcome, and only holding together in the forward direction as a side effect of its ultimate convergence.
So you would still need a stronger version of your last admitted thesis that not only establishes physical causation of the mental, but physical causation of the physical.