Is the Planck length Lorentz invariant?

A possible answer to the last part of the question: the article Six Easy Roads to the Planck Scale, Adler, Am. J. Phys., 78, 925 (2010) contains multiple "derivations" that you might (or might not) find more satisfactory than the one you mention.

As far as the rest of the question is concerned, others have made the most relevant points. I think a fair summary of what Magueijo is getting at is something like the following:

One frequently hears that "interesting new physics" happens when some length $l$ is less than the Planck length. The Planck length is manifestly Lorentz invariant. The other length $l$, if it is the physical length of some object, is manifestly not Lorentz invariant. What meaning, then, can one assign to such statements?

It seems to me that reasonable people can differ over whether this is an interesting question. I don't find it manifestly insane, myself.

I don't know if anyone is still watching this thread, but anyway, the 2001 paper referred to by the OP describes an idea called doubly special relativity (DSR). There is a WP article on it, which provides a more current view. Basically my impression is that DSR didn't work out well, and nobody, including Magueijo and Smolin, is really working on it anymore.

For an answer to this question in the context of loop quantum gravity, see Rovelli and Speziale, "Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction," http://arxiv.org/abs/gr-qc/0205108 .

The OP asked the question in a somewhat naive way, but that doesn't mean that the whole issue is trivial. In SR, we have a constant called c. It's constant by definition. But that doesn't mean that it's a trivial statement that when an observer sees a particle as having a velocity equal to c, that fact can be Lorentz invariant.

To answer this question one should first define the Planck scale operatively, i.e. to define how different Lorentz observers would measure it in (perhaps thought) experiments.

Usually one defines coupling constants in terms of local, (quasi) static experiments that each observer can perform in its rest frame. Thus G, for example, is measured in such experiment and therefore is, by definition, Lorentz invariant, in spite of being a dimensionful quantity.

c on the other hand clearly cannot be measured that way, but it is not a coupling constant, but a dimensionful observer-independent scale used to define what Lorentz transformations are (Lorentz transformations are such linear spacetime transformations that allow for the presence of an invariant velocity scale c).