Yeah … maybe something like the following would work:
(1) estimate the total shots N, via Poiss(lam_tot)
(2) estimate the the probabilities (p1, p2, p3) of high, med, low via a Dirichlet distribution.*
*note: Here (p1, p2, p3) would be close to the league average, but there would be a fixed amount of variation, say sigma, which represents how much each team can vary around the mean … (that is, this would be a hieracrghical model if, say. sigma in HN(0.2) is also estimated as a parameter)
(3) Plug N and (p1, p2, p3) into a multinomial distribution (rather than binomial, because more than 2 probabilities)
(4) sampling k1, k2, k3 ~ Multinomial(p1,p2,p3, N) we get counts k1, k2, k3 such that Sum kj = N which are estimates on the number of high, med, low danger shots …
(5) Then, at each level, one could repeat this process … for example, if you wanted to split the number of high danger shots into, 5 new sub-categories k1 —> k11, k12, k13, k14, k15 … one could estimate probabilities (p11, … p15) and so on …
side note: Now, I believe if you let N vary over Poiss(lam_total), then sampling over bith distributions
k1, k2, k3 ~ Multinomial(p1,p2,p3, N), N ~ Poiss(lam_tot)
is equivalent to selecting from several Poisson distributions kj ~ Poiss(lam_j = pj lam_tot) … in each case Sum kj = N, but the N varies from sample to sample …