Laplace's daemon in absence of global simultaneity

The qualification 'no meaning' is too strong.

Minkowski spacetime does feature relativity of simultaneity, but this relativity of simultaneity is itself subject to physical law.

In a spacetime diagram one can specify a plane of simultaneity. Depending on the choice coordinate system (among the equivalence class of inertial coordinate systems) the orientation of the plane of simultaneity comes out differently; this difference is perfectly coordinated.

In terms of newtonian dynamics the members of the equivalence class of inertial coordinate systems are related by galilean transformation. In terms of special relativity the members of the equivalence class of inertial coordinate systems are related by Lorentz transformation.

The crucial property is that the members of the equivalence class of inertial coordinate systems are in a coordinated relation with respect to each other.

One should think of the equivalence class of inertial coordinate systems as a coordinated superstructure.

That is why in terms of Minkowski spacetime it is still possible for Laplace's daemon to have access to an exhaustive specification of the motion state of the entire universe.

Coming from the general relativity angle, the issue is whether there exist a Cauchy hypersurface that ensures predictability. Spoiler: it does, and the daemon can do its job.

A Cauchy hypersurface is a subset of space-time which is intersected by every inextensible, non-spacelike (i.e., causal) curve exactly once. If they exist then there is a homeomorphism from the spacetime manifold $M$ to $S times mathbb{R}$ where $S$ is the 3D Cauchy hypersurface (Geroch 1970). This means that spacetime can be foliated by a progression of spacelike hypersurfaces ("moments") stacked along some form of well-defined time direction. This is called global hyperbolicity.

Global hyperbolicity ensures predictability. A system is "predictable" if its state on a Cauchy surface uniquely determines its state at any future point. A physical theory is "prognostic" if all systems described by the theory are predictable. Theories such as relativistic mechanics and electrodynamics are prognostic theories on globally hyperbolic manifolds. For globally hyperbolic spacetimes and predictable matter fields general relativity is prognostic: the metric of spacetime is uniquely determined by the field equations and knowing it on a Cauchy surface (Hawking & Ellis 1973).

The payoff for all this technical stuff is that it looks like that at least classical mechanics in a general relativity setting is prognostic and predictable. Laplace's daemon will be able to do its job if it just has all the data on one Cauchy surface - this is the data it needs. Doing this on the special case of flat Minkowski space like the original question asks is particularly easy, since we know they exist: just take anybody's personal $t=0$ surface.

In that case how to think of the Laplace's daemon knowledge at a specific moment?

Laplace's demon has knowledge of momentum and position of every particle in the universe at a specific moment in time IN ITS FRAME. And since it knows this, and it has infinite computational power, it can calculate the momentum and position of every particle in any other frame at any moment in time.