Must the action be a coordinate scalar?

I know that an action must be locally-Lorentz invariant based on physical reasons, but is there any requirement for it to be a coordinate pseudo-scalar (up to surface terms)? In particular, would an action of the following form be permissible,
$$
S=int sqrt{-g} L , d^4x ,
$$
where $L$ is not a coordinate scalar (i.e. has no free indices but explicitly depends on coordinates) but is still a local Lorentz scalar. An example of such a Lagrangian that isn’t a coordinate scalar would be using a contraction of the Christoffel symbols
$$
L= g^{mu nu}Gamma_{mu lambda}^{rho} Gamma^{lambda}_{nu rho} .
$$

Will this still give a well-defined variation principle $delta S = 0$ despite $S$ being non-covariant and depending on the choice of coordinates?

A similar question but regarding Lorentz invariance is asked here Must the action be a Lorentz scalar? but I’m unsure if the argument also applies here too (with diffeomorphism invariance instead of Lorentz invariance).

Also note I do not care if the equations of motion resulting from $delta S = 0$ are not covariant.

—
On further thought, rather than Lagrangian wrote above one could just consider something simple like
$$
L = partial_{mu}partial_{nu}g^{mu nu}
$$
which is probably easier to work with. One potential problem I see is that the action could vanish or diverge to plus or minus infinity in some coordinate systems – is this a problem for the variation principle $delta S= 0$?