The planck length is defined as $l_P = \sqrt{\frac{\hbar G}{c^3}}$. So it is a combination of the constants $c, h, G$ which I believe are all Lorentz invariants. So I think the Planck length should also be a Lorentz invariant! But there seem to be some confusion about that, see e.g. the following paper Magueijo 2001: Lorentz invariance with an invariant energy scale:
The combination of gravity $G$, the quantum $h$ and relativity $c$ gives rise to the Planck length, $l_p$ or its inverse, the Planck energy $E_p$ . These scales mark thresholds beyond which the old description of spacetime breaks down and qualitatively new phenomena are expected to appear. ... This gives rise immediately to a simple question: in whose reference frame are $l_P$ and $E_P$ the thresholds for new phenomena?
But if $l_P$ is a Lorentz invariant their is no question about that. $l_P$ is the same in all reference frames! Another confusing issue is that the Planck mass (from which the Planck length is derived) is often derived by setting equal the Compton length $\lambda_C = \frac{h}{m_0 c}$ ( a Lorentz invariant 4-length) and the Schwarzschild length $r_{s} = \frac{2Gm}{c^2}$ (which I believe is not a Lorentz invariant, since in the derivation of the Schwarzschild metric it is assumed to be a 3-length, measuring a space distance). But since Compton wavelength and Schwarzschild radius are not lengths of the same kind I think such a derivation is not valid. So my question is:
Is the Planck length a Lorentz invariant and if so, how to derive it then without using the Compton wave length and the Schwarzschild radius ?