Read through interval scheduling with eligible machine sets (lax-888481), the zoo problem $P \mid M_j \mid \sum w_j Z_j$ inherits the two hardness results of that problem.

W[1]-hardness for the number of machines. Multicoloured Clique, parameterized by the number of colours, fpt-reduces to interval scheduling parameterized by the number of machines (Theorem 1 of lax-888481), and that problem fpt-reduces to $P \mid M_j \mid \sum w_j Z_j$ parameterized by the number of machines, by padding. Both reductions are strict fpt-reductions of the archive, computed by word RAM programs in linear fixed-parameter time; they raise the parameter by computable functions of it, and the numbers they write are below $2^{e(k)\,(|x|+1)}$ for a computable $e$, where $|x|$ is the bit size of the input and $k$ its parameter. So both are fpt-reductions in the sense of lax-496464, and so is their composition.

Theorem. Multicoloured Clique fpt-reduces to $P \mid M_j \mid \sum w_j Z_j$ parameterized by the number of machines. Since Multicoloured Clique is W[1]-complete (lax-496464), the problem is W[1]-hard.

NP-hardness for $p_{\max} \le 25$. For an interval scheduling instance $I$, let $I'$ be its padding: a job without an eligible machine gets one new machine and weight $0$. Read $I'$ as an instance of $P \mid M_j \mid \sum w_j Z_j$ with threshold $\sum_j w'_j - \sum_j w_j$, the total weight of $I'$ minus the total weight of $I$.

Theorem. Deciding this instance is NP-hard already on the interval scheduling instances with processing times at most $25$ and unit weights: every language in NP reduces to it by a polynomial-time Turing machine writing the encoding of $I$. On such an instance the threshold is met exactly when every job of $I$ can be scheduled, which is NP-hard by Theorem 2 of lax-888481. Padding keeps the processing times, so the zoo instances have $p_{\max} \le 25$, and the problem is para-NP-hard for $p_{\max}$.