The thing is, this should work, but it likely won't.
You'll find that doing well in courses and research while also working is not feasible in 3 to 4 years, maybe you could do it in 5 or 6 years. Even then, you'll find community college classes and an online master's will be discounted. You'll also find that the instructors in these online classes might be grad students or not active researchers, meaning their letters are not very valuable.
If your goals do not include academia, you might find that doing an MPP (in person, not online) and then a PhD in Public Policy can get you where you want to go at much lower cost.