There has been some progress on this question since the question was asked. Apparently it was "known to the experts" that there cannot be an integral $p$-adic cohomology theory which is finitely generated, agrees with rigid cohomology rationally and satisfies finite étale descent. This was mentioned by Richard Crew at least at the 2017 conference "$p$-adic cohomology and arithmetic applications" at BIRS. The complete argument (failure of descent along Artin-Schreier covers) is very nicely written up in [AC22].
If one assumes some strong form of resolution of singularities in positive characteristic, then [ESS23] shows that there is a "good" (well-defined, functorial, finitely generated, rationally agrees with rigid cohomology,...) integral $p$-adic cohomology theory which satisfies $\mathrm{cdh}$-descent. It is given by sending a variety $X$ to the logarithmic crystalline cohomology of an appropriate normal crossings compactification $(\overline{X},\overline{X}\setminus X$)$.
Without assuming resolution of singularities, there is [Mer24] which proves that a "good" integral $p$-adic cohomology theory satisfying Nisnevich descent exists. If one assumes some version of resolution of singularities, it is shown that this theory agrees with the cohomology theory in [ESS23].
As far as I know, none of this has been worked out with coefficients yet.
[AC22] T. Abe, R. Crew, Integral $p$-adic cohomology theories, 2022, arXiv:2108.07608v2
[ESS23] V. Ertl, A. Shiho, J. Sprang, Integral $p$-adic cohomology theories for open and singular varieties, 2023, arXiv:2105.11009v2
[Mer24] A. Merici, A motivic integral $p$-adic cohomology, 2024, arXiv:2211.14303v4