Can an employer(a government entity) require my mandatory participation in an organization that lobbies for that same employer (a government...

Question

Can an employer(a government entity) require my mandatory participation in an organization that lobbies for that same employer (a government entity) under threat of adverse action for non-participation?

Answer

Since you chose to omit all the details that matter (which government entity - state, federal, local, etc), what specific organization, and what adverse action, all of which are central to the answer there is no way to answer you.