There are several aspects to your question, and I think that asking for a cocycle of transition functions is partly misleading. The point here is that as you observe in the quesiton, you can define $\mathcal E(1)$ as a density bundle, without making reference to a projective structure. Thus also $J^1\mathcal E(1)$ is, as a vector bundle, independent of the projective structure. Likewise, the canonical Cartan bundle associated to a projective structure is, as a principal fiber bundle, independent of the projective strucutre. (You can define it as $\mathcal PM\times_{GL(n,\mathbb R)}P$, i.e. by an extension of structure group of the linear frame bundle of $M$.) The actual projective structure is not encoded in these bundles but in the canonical linear connection on $\mathcal T^*$ or equivalently the canonical Cartan connection.

To understand $\mathcal T^*$ as a vector bundle, recall that there is a canonical exact sequence $T^*M\otimes\mathcal E(1)\to J^1\mathcal E(1)\to \mathcal E(1)$ (the jet exact sequence). This alrady contains all the information needed to see that it is naturally associated to the representation $\mathbb R^{(n+1)*}$ of $P$. Since $P$ is the stabilizer of a line in $\mathbb R^{n+1}$ it can be equivalently realized as the stabilizer of a hyperplane in $\mathbb R^{(n+1)*}$. (The hyperplane is the annihilator of the distinguished line.) Being associated in this sense just means containing a distinguished subbundle of hyperplanes.

Now you can further recall that choosing a linear connection on $\mathcal E(1)$ defines a splitting of the jet exact sequence, so as a vector bundle $J^1\mathcal E(1)\cong (T^*M\otimes \mathcal E(1))\oplus\mathcal E(1)$, which tells you how to write down transition functions from those of $T^*M$ and $\mathcal E(1)$. For any projective class $[\nabla]$ mapping a connection on $T^*M$ to the induced connection on $\mathcal E(1)$ defines an isomorphism between connections in the projective class and connections on $\mathcal E(1)$, which leads to the standard picture of splittings of $\mathcal T^*$ associated to the connections in the projective class.

The reason why I consider the question of transition functions as partly misleading is that the description of $\mathcal T^*$ as $J^1\mathcal E(1)$ also tells you that diffeomorphisms of $M$ act on $\mathcal T^*$ in a specific way. This information easily gets lost if you describe the bundle in terms of transition functions. The "right" action is given as follows: A diffeomorphism $f:M\to M$ induces a pullback operator $f^*$ on sections of $\mathcal E(1)$, basically given by composing with $f$ and multiplying by an appropriate power of $|det(Df)|$. As you can see, $f^*\sigma(x)$ alredy depends on first derivatives of $f$ in $x$. The natural action on $J^1\mathcal E(1)$ then basically maps $j^1_x\sigma$ to $j^1_x(f^*\sigma)$, so this depends on second derivatives of $f$ in $x$. Hence $\mathcal T^*$ is not associated to the frame bundle of $M$ but only to the second order frame bundle. (Things can be similarly phrased in terms of the Cartan bundle.) With the "right" action of diffeomorphisms, it is then true that a diffeomorphism preserves the projective class (which singles out a finite dimensional subgroup of the diffeomorphism group) if and only if the induced map on $\mathcal T^*$ is compatible with the tractor connection.