Marginalizing in simplified discrete model

Modeling
1

These parameters aren’t identified. I want to marginalize out u to see if that helps because it helped in an earlier problem I had. Is that doable here, in the more discrete model?

data {
  int<lower=0> N;
  int<lower=1> K;
  int y[N];
  int kk[N];
}
parameters {
  vector[N] u;
  real<lower=0> mu[K];
  real<lower=0> sigma;
  real<lower=0> offset[K];
  vector[K] g0;
}
transformed parameters {
  real beta[K];
  for (k in 1:K) {
    beta[k] = offset[k] * sigma + mu[k];
  }
}
model {
  mu ~ normal(1, 3);
  sigma ~ normal(1,1);
  to_vector(g0) ~ normal(0,3);
  to_vector(u) ~ normal(0,1);

  for (n in 1:N) {
    kk[n] ~ categorical_logit(u[n]*g0);
    y[n] ~ bernoulli_logit(u[n]*beta[kk[n]]);
  }
}