Math preparedness for an Econ PhD
This type of preparation would be required for both an economics PhD and an econ-focused Public Policy PhD (one with a substantial number of courses in Econ department or co-listed). You would probably get admitted to a Public Policy PhD without real analysis, but you would then struggle in the courses if they were taken in the Econ department. Another area you would want to focus on is proof-writing.

I would suggest taking these courses at a 4-year institution if at all possible. You want the grades you receive to be reliable signals of your learning. The goal is to give the admissions committee confidence that you will succeed in the coursework. Many Econ and some Public Policy programs will force you to leave at the end of year one without satisfactory performance on qualifying exams.

You are correct that you can do similar work in both degree programs. In an Econ department, using the most flashy new methods is weighted more heavily, while in the Public Policy department, useful questions and answers are more heavily weighted. But if you find a dissertation topic with the right combination of questions and methods it would likely be acceptable in either degree program.

You are also correct that the Economics degree is more valued on the job market and will earn you a higher salary for the same position. I have a Public Policy friend who was told that he would have received a $10K higher starting salary if his degree had the word Economics in the name.