Your question presumes a variety of framing assumptions which are now generally held not to be true by most philosophers.
First, you are presuming that "completeness of physics" would mean that "physics is everything", IE a reduction of all thought to physics, including the philosophy of science. But reductionism is no longer considered viable in most science, and certainly not OUTSIDE science, in philosophy of science! https://plato.stanford.edu/entries/scientific-reduction/
This point was very effectively demonstrated when the Logical Positivists tried to subsume philosophy into science using the Verification Principle. The VP itself is meaningless per its own terms, https://www.newworldencyclopedia.org/entry/Verifiability_principle The point of this failure is that science is part of a suite of thought disciples -- IE pluralism of valid modes of thinking applies not just within the sciences, but across all fields of thought. And notably, philosophy in the form of epistemology and logic, are both preconditions for science to even operate. Therefore physics cannot ever be logically closed.
Additionally, a dominant movement in philosophy is analyticity -- the premier role of logic in philosophic thinking. Analytics depend on precise definitions, hence the problem you face with the definition of physics, and its fuzzy boundary where science is still being done. Quine, in Two Dogmas of Empiricism, showed that NO language will ever be precise enough to support analyticity https://www.theologie.uzh.ch/dam/jcr:ffffffff-fbd6-1538-0000-000070cf64bc/Quine51.pdf by inference, the very fuzzy nature of the boundaries of physics (or of science) is even less precise than language, so your effort to precisely define the terms of physicalism in terms of physics is doomed to failure. For reference, this book explored the definitional approach to physicalism, and found it to be fruitless: https://www.amazon.com/Physicalism-Problems-Philosophy-Daniel-Stoljar/dp/0415452635/ref=cm_cr_srp_d_product_top?ie=UTF8 (Notably, Stoljar concluded that physicalism was actually refuted by the peculiarities of modern physics).
Hempel's Dilemma only applies to this definitional/analytic/rational approach to physicalism. A potentially more fruitful approach is to treat physicalism as an empirical "hypothesis". In empiricism, definitions are not as critical -- they can be adjusted on the fly as one does further investigations. An excellent generalization of how to do empiricism is found in the Research Programme language of Imre Lakatos. https://antimatter.ie/2011/02/11/kuhn-vs-popper-the-philosophy-of-lakatos/ And one defender of physicalism has cast it explicitly in Lakatosian terms: https://core.ac.uk/download/pdf/74162.pdf
Papineau's paper explains very clearly why physicalism became a dominant view in philosophy in the 20th century. It also implies why physicalism is struggling today. Hempel's Dilemma is but one minor problem. For physicalism to be a global worldview, everything has to reduce to physics, and reductionism has failed as a project, both in science, and across knowledge generally. When science relies upon philosophy (epistemology and logic) and on math, and neither of these fields reduce to physics -- the claim that physics is everything there is -- is pretty explicitly self-refuted! A plurality of philosophers hold by both abstract object realism, and moral realism, both of which are effectively incompatible with physicalism. Plus there is widespread consensus that the "hard problem of consciousness" has not been solved, and there are no current promising routes to solve it -- which leaves consciousness too as a major unaddressed subject outside physics. https://www.amazon.com/Physicalism-Something-Princeton-Monographs-Philosophy/dp/0691133859 Worse, most theoretical physicists actually appear to be IDEALISTS -- IE they consider matter to be a reflection of MATH. So even if reductionism were somehow salvageable -- reduction itself is now problematic for physicalism.
So, as an answer to your question, no, you cannot solve Hempel's Dilemma definitionally, nor can you include epistemology in this redefinition of physics. And the problems physicalism faces are much worse than just Hempel's Dilemma.