$\newcommand{\RP}{\mathbb{RP}}\newcommand{\C}{\mathbb C}\newcommand{\cC}{\mathcal C}$Here's a TFT-style argument
for why it should be possible in principle to use an invariant of triangulations to distinguish $\RP^4$ from
Capell-Shaneson's fake $\RP^4$, which I'll call $Q$; however, the specific invariant needed has likely not been
constructed. (Moishe Kohan's comment is a much faster argument that such a combinatorial invariant exists, but hopefully this answer makes it more explicit what it would look like.)

Given a general $n$-dimensional pin^{+} TFT $Z'\colon\mathsf{Bord}_n(\mathrm{Pin}^+)\to\cC$, and
for a nice choice of target category $\cC$, there is expected to be an $n$-dimensional unoriented TFT
$Z\colon\mathsf{Bord}_n\to\cC$ obtained by “summing over pin^{+} structures,” akin to
the finite path integral in Dijkgraaf-Witten theory. For example, if $M$ is a closed, unoriented $n$-manifold and
$P^+(M)$ denotes its set of pin^{+} structures,

$$ Z(M) = \sum_{\mathfrak p\in P^+(M)} \frac{Z'(M, \mathfrak p)}{\#\mathrm{Aut}(\mathfrak p)}.$$

If $Z'$ is fully extended, and $\cC$ is chosen appropriately, it should be possible to define $Z$ as a fully
extended TFT as well. At present, though, I think this has only been shown up to category number 2 (once-extended
TFTs).

Moreover, it's believed that fully extended TFTs (again, for certain choices of target category $\cC$) can
all be constructed using state sums, with input data of a triangulation. There is work of Kevin Walker on
implementing this, though I don't know exactly what assumptions (e.g. choice of $\cC$) he works with.

Let's use this strategy to build a 4d unoriented TFT $Z$ which distinguishes $\RP^4$ from $Q$. Let $\zeta :=
e^{i\pi/8}$ and $\mu_{16}\subset\C^\times$ denote the multiplicative group of 16^{th} roots of
unity, which is generated by $\zeta$. The 4d pin^{+} $\eta$-invariant is a $\mu_{16}$-valued invariant of
the Dirac operator on a pin^{+} 4-manifold; for the two pin^{+} structures on $\RP^4$, it takes
on the values $\zeta^{\pm 1}$, and for the two pin^{+} structures on $Q$, it takes on the values
$\zeta^{\pm 9}$. This is discussed in Kirby-Taylor, “Pin structures on low-dimensional
manifolds”; they also show this
$\eta$-invariant is a pin^{+} bordism invariant.

Freed-Hopkins show that any $\mathrm U_1$-valued bordism invariant $\alpha$ lifts to an invertible TFT $Z'$ such
that in top dimension, $Z'(M) = \alpha(M)$. Such a TFT is expected to be fully extended, but so far has only been
constructed down to codimension 2, with target 2-category the Morita category of superalgebras over $\C$. In
any case, applying this to the $\eta$-invariant produces a 4d pin^{+} TFT, which will be our $Z'$.
Summing over pin^{+} structures as above, we obtain a 4d unoriented TFT $Z$, with values

$$ Z(\RP^4) = \frac{\zeta + \zeta^{-1}}{2},\qquad\quad Z(Q) = \frac{\zeta^9 + \zeta^{-9}}{2}.$$

Thus $Z(\RP^4)$ is a positive real number and $Z(Q)$ is a negative real number, so we have an (in principle) fully
extended 4d unoriented TFT distinguishing $\RP^4$ and $Q$, hence which should admit a state-sum description.

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