IV-type estimator for multinomial endogenous treatment and multinomial outcome?

I am modeling the effect, among other things, of health insurance on choice of "health" provider consulted in developing country X. I want to fit the following:

Provtype_i=b₀ + b₁Insur1_i + b₂Insur2_i + b₃Insur3_i + Xb + e_i

Where Provtypei = Type of provider chosen (No care (Ref.); Traditional healer; Public sector; Private sector) and Insur#_i are health insurance dummies as follows:

* Insur1_i = Employer-based health insurance (government-run health insurance for civil service and formal private sector employees)--8.7% of sample
* Insur2_i = Indigent health insurance--11% of sample
* Insur3_i = Other type of health insurance-- 4.2% of sample
* "No insurance" is the reference category and the largest (~76% do not have health insurance)

Unweighted analytic sample size is 5,014. When weighting is used, it is 4,175.

Literature suggests health insurance enrolment is potentially endogenous. I have several instruments based on the literature, but I am having a difficult time imagining how to model this in Stata. The only commands I've seen for multinomial endogenous treatment are -mtreatreg- and -mtreatnb-. Is -gsem- (with mlogit link) the only way to go here? How can I "tell" Stata that the same unobserved variable affects all 3 insurance choice?