I agree with @PaigeMiller that you are on the right track.
I agree with @PGStats that you need to consider whether to exclude other potential interactions among the fixed effects (and be able to justify their exclusion).
In addition, I would think of this as an ANOVA rather than a regression. Of course ANOVA is really just a special case of regression, but there are things we can apply in ANOVA models that we don't apply in regression (like LSMEANS as @PaigeMiller suggested). ANOVA models have assumptions (notably, normality and homogeneity of variance and independence) that you would need to assess for your analysis. SAS® for Linear Models, Fourth Edition is an excellent resource; these are topics that are also covered extensively in courses, and so there are resources available on the internet, in books, etc.
I hope this helps.
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