The only physical constant I know for sure is Lorentz-Invariant is electric charge. I am curious to know if there are others even if it's not possible to make an exhaustive list.
All physical constants are “invariant” in the sense that all observers agree on their values. But we do not call things like Planck’s constant or Boltzmann’s constant “Lorentz invariants”.
Here are some examples besides charge of Lorentz-invariant quantities that are dynamical, kinematical, or geometrical:
The speed of light. It’s the invariant magnitude of every object’s four-velocity.
Masses of elementary particles (and more complicated systems). Mass is the invariant magnitude of the energy-momentum four-vector. (I hope that you did not get taught the confusing and obsolete concept of “relativistic mass”!)
The spacetime interval between any two points in Minkowski spacetime.
The difference of the squares of the electric and magnetic field strengths at any point in spacetime.
The value of the Higgs field at any point in spacetime.
The scalar product of any two four-vectors.
As you can see, there are lots of examples.
The best way to understand what transforms and what doesn’t under Lorentz transformations is to learn physics written in “manifestly covariant” form using four-vectors, four-tensors, etc. If a quantity doesn’t have an index, or if all indices in an expression are contracted, then it is manifestly a Lorentz invariant.