A good alternative to the global Euler characteristic of Tate is then $$ \chi_{c,U}(F)=\prod_i [H^i_c(U,F)]^{(-1)^i} $$ It turns out that $\chi_{c,U}$ extends to an Euler characteristic $\chi_{W,U}$, the Weil-étale Euler characteristic, valued in $\mathbb{Q}_{>0}$ and defined for all $\mathbb{Z}$-constructible sheaves on $U$.

Now from a formal dévissage argument using explicit computations for sheaves supported on closed points and the theorem, it follows that $\chi_{W,U}(F)=\chi_{c,U}(F)=1$ for a constructible sheaf $F$. In my view, this is the right generalization. Note that if $F$ is constructible locally constant, corresponding to a $G_{K,S}$-module $M$ such that $l$ is invertible on $U$ for each $l$ dividing the order of $M$, then $H^i(U,F)=H^i(G_{K,S},M)$ and the statement $\chi_{c,U}(F)=1$ is equivalent, after unwinding the definitions, to the formula for the global Euler characteristic (thus by a dévissage one can also prove that $\chi_{c,U}(F)=1$ for $F$ constructible by reducing to the above case and the case of a sheaf supported on a closed point). When the invertibility condition does not hold the relation between $H^\ast(U,F)$ and $H^\ast(G_{K,S},M)$ is more complicated: let $\bar{U}$ denote the normalization of $U$ in the fixed field of $G_{K,S}$. Then you have a spectral sequence $$ H^p(G_{K,S},H^q(\bar{U},F))\Rightarrow H^{p+q}(U,F) $$

Side note: everything I said is studied in my article Special values of L -functions on regular arithmetic schemes of dimension 1. I apologize for the self-plug but I feel like it was kind of inevitable

A good alternative to the global Euler characteristic of Tate is then $$ \chi_{c,U}(F)=\prod_i [H^i_c(U,F)]^{(-1)^i} $$ where $[-]$ denotes cardinality. It turns out that $\chi_{c,U}$ extends to an Euler characteristic $\chi_{W,U}$, the Weil-étale Euler characteristic, valued in $\mathbb{Q}_{>0}$ and defined for all $\mathbb{Z}$-constructible sheaves on $U$.

Now from a formal dévissage argument using explicit computations for sheaves supported on closed points and the theorem, it follows that $\chi_{W,U}(F)=\chi_{c,U}(F)=1$ for a constructible sheaf $F$. In my view, this is the right generalization. Note that if $F$ is constructible locally constant, corresponding to a $G_{K,S}$-module $M$ such that $l$ is invertible on $U$ for each $l$ dividing the order of $M$, then $H^i(U,F)=H^i(G_{K,S},M)$ and the statement $\chi_{c,U}(F)=1$ is equivalent, after unwinding the definitions, to the formula for the global Euler characteristic (thus by a dévissage one can also prove that $\chi_{c,U}(F)=1$ for $F$ constructible by reducing to the above case and the case of a sheaf supported on a closed point).

Since the groups $H^i_c(U,F)$ are finite, the groups $H^i(U,F)$ are also finite because Galois cohomology of $\mathbb{R}$ and of p-adic fields/henselian local fields of number fields with finite coefficients is finite. We can thus define for $F$ constructible: $$ \chi_U(F)=\prod_{i=0}^3 [H^i(U,F)]^{(-1)^i} $$ and the equality $\chi_c(U,F)=1$ becomes: $$ \chi_U(F)=\prod_{v\in S} \prod_{i=0}^3[H^i(K_v,F_v)]^{(-1)^i}=\prod_{v~\text{finite}} \chi_v(F_v) \prod_{v~\text{archimedean}} \frac{[H^0(K_v,F_v)]}{[\hat{H}^0(K_v,F_v)]} $$ where $\hat{H}^i(K_v,-)$ for $v$ archimedean denotes Tate cohomology and $\chi_v$ is the local Euler characteristic (I have used that the Herbrand quotient of a finite module is $1$). Using the computation of the local Euler characteristic [1, I.2.8] and the product formula, we find $$ \chi_U(F)= \prod_{v~\text{archimedean}} \frac{[H^0(K_v,F_v)]}{[\hat{H}^0(K_v,F_v)]|[M]|_v} $$ where $M$ is the $G_K$-module corresponding to the stalk $F_\eta$ at the generic point and $|-|_v$ is the normalized norm associated to the archimedean place $v$. In the case where $mF=0$ for $m$ invertible on $U$ this is [1, II.2.13].

When the invertibility condition for a locally constant constructible sheaf $F$ does not hold, the relation between $H^\ast(U,F)$ and $H^\ast(G_{K,S},M)$ is more complicated (and I don't even know if the latter are finite!): let $\bar{U}$ denote the normalization of $U$ in the fixed field of $G_{K,S}$. Then you have a spectral sequence $$ H^p(G_{K,S},H^q(\bar{U},F))\Rightarrow H^{p+q}(U,F) $$ One can say that $H^0(\bar{U},F)=M$, $H^1(\bar{U},F)=0$ and $H^i(\bar{U},F)(l)=0$ for $i\geq 0$ if $l$ is invertible on $U$ (see [1, the proof of I.2.9]); beyond that I don't know what can be said.

Side note: a lot of what I said is studied in my article Special values of L -functions on regular arithmetic schemes of dimension 1. I apologize for the self-plug but I feel like it was kind of inevitable