What typically one would do would be to look at the fit statistics overall for the Latent Class run. You will get a Log-Likelihood for the Latent Class model. You can compare the LL for the latent class models with main effects only vs. the LL for the result when you include interactions. (Make sure to keep the number of classes constant between the models).
I may be wrong here, but the statistical test would involve counting the number of parameters added to the model times the number of segments in your LC solultion (somebody please correct me if I'm wrong).
So, if you were adding a 2-way interaction between an attribute with 5 levels and another with 4, there would be an additional (5-1)(4-1)=12 parameters to the model. But, since we have three groups, I think it means there are actually 12*3 = 36 degrees of freedom for our statistical test (the 2-LL test).
So, if you saw a difference in LL of 10 points between the model with and without interactions, then the Chi-square test involves a critical value of 10x2 = 20 with 36 degrees of freedom. Use the =ChiDist(20,36) formula in Excel to compute the p-value. A p-value of 0.05 or less would indicate that the interaction effect was adding significant fit.