My understanding of the Lorentz transformation is that to ensure that laws of physics remain frame-independent, a transformation was devised, which we call today by the name Lorentz Transformation.
But how do we know that there does not exist any other transformation which ensures that the laws remain frame independent, or that Lorentz Transformation is a special case for some other transformation?
Assuming
gives the Lorentz transformations with a free parameter C which specifies a maximum speed. If C is infinite you get the Galilean group, while for C finite you get the usual Lorentz transformations. (C is later identified physically with the speed of light in vacuum if you accept that the photon is massless).
So there are only two possibilities under those above four assumptions. The derivation has been rediscovered many times since Ignatowski did something similar it in 1910, see eg http://o.castera.free.fr/pdf/One_more_derivation.pdf
If you relax any one of the assumptions above you get generalisations which might be useful when searching for potential deviations from special relativity. See eg https://arxiv.org/abs/1302.5989
The answer to your question depends on exactly which laws of physics you want your transformations to preserve, and it depends also on how you define the phrase "Lorentz transformation".
For example, if you require your Lorentz transformations to be linear, then they are not the only transformations that preserve the light cone. Another such transformation is: $$(x',t')=(x,t) \hbox{ if $x$ and $t$ are both rational}$$ $$(x',t')=({x-t\over2},{x+t\over 2}) \hbox{ otherwise}$$
Does this transformation count as one that "preserves the laws of physics"? We can't know until you specify exactly which laws you want preserved.
There's another related issue: Although you asked about individual transformations, what you are really most likely to be interested in is the function that associates to each velocity a transformation $L_v$. In addition to asking about the properties of the allowable individual transformations, you are likely to also be interested in the properties of this function. For example, you're very likely to want $L_{v^{-1}}=L_v$ for all $v$. You also might want the assignment $v\mapsto L_v$ to be continuous or differentiable. These conditions can end up restricting the possible values for $L_v$.
If you require each $L_v$ to be a differentiable function of $x$ and $t$, and if you require the assignment $v\mapsto L_v$ also to be differentiable, and if you impose some natural symmetries, and if you require that the order of events be preserved, then you ought to be able to show (just by differentiating the obvious expressions and manipulating) that $L_v$ must be linear (and hence an element of the usual Lorentz group). You also might be able to drop some of these assumptions and replace them with other assumptions that have the flavor of "such and such a law of physics must be preserved". But again, that's going to depend on exactly which laws you have in mind, and unless/until you state them unambiguously, there can be no unambiguous answer to your question.
