Hi! Well, I have some perspective on phd programs in econ and experience in the public sector that may be helpful to you.
First, I would say, yes, you can do anything you put your mind to, if you are willing to make the necessary sacrifices.
Second, I would say that you would be just fine with an MA in the public sector, and a PhD is not at all a requirement. I worked in a large government agency in which people with MAs did just as well as those with PhDs. Granted, it might take a few more years, but those with PhDs spent a few more years in school. MAs in econ have pretty good marketability -- I know because I have one. You can enter at a decent level and move up in research consulting firms, think tanks, and the public sector. There may be a bit of a glass ceiling in some of those places (not the public sector though) for those without a PhD, but there are always cases of those who break through.
So, I mentioned I have an MA in econ. I studied at a top 25 or so ranked school about 10 years ago. Let me just say that grad school in economics is extremely brutal. You may or may not already know that, but most programs eliminate a 10th to third of the class through comprehensive exams. During the first year, you struggle to keep up with your classmates, who are the best and the brightest in the WORLD because so many international students study econ in the U.S. Everyone pretty much feels like they are about to fail out, because it is just that hard. After each exam, where you are put through the meat grinder, you have to pick yourself back up and get ready for the next blow. The second year is not much easier in terms of time management. After that, you are in dissertation phase, if you make it that far. At that point, if you have supportive faculty, then it can be more manageable.
So, I did not make it through the comps. I left with an MA. It was brutal and horrific, I barely made it through the first year psychologically speaking, and it was a tremendously difficult experience. Indeed, I had no problem at all finding a job, but the feeling of failure was just awful. I watched my former classmates go on to move into higher positions than me at the same research firm. I felt I could not have autonomy and a renewed sense of self-confidence without a phd.
So, I went back for a PhD. Not in my own field but in a related one. I had my first child in the first year and my second in my last year. It was totally doable in this program. But, here I am at the end of it all with a PhD and my job prospects are more bleak than they were before. I am trained to be an assistant professor, which earns a salary about 75 percent of what I was making before I went back. I am over-qualified for data analysis jobs which might have permitted part-time work, which I really wish I could do to spend more time with the kids. It is quite a conundrum.
So, I totally get that a BA in econ is not enough. An MA in something is definitely worth it. However, think very carefully about whether that something should be econ. The benefit of econ is that they often cover your studies with fellowships or assistantships, so you don't leave with a ton of debt the way you do with public policy. But it is very, very hard to not buy into the idea of getting a phd like everyone else in the program and "just leave with a Masters." At the same time, PhDs in econ have decent job marketability. If you take a job as assistant professor, the salary may be a little low, but there may be opportunities later on for consulting. If you take a nonacademic job, you are looking at starting salaries of about $90K. But, all of this is assuming you want full-time work, of course.
So, you are a single mom already, and need earning power. But, you also need to be able to make it through whatever program you choose. Econ is not a family friendly program whatsoever. You will need to study all the time, and it will be very hard to develop the comraderie with other students when you need to be home for your child. I'm quite certain I could not have made it through the first and second years of my econ program with a young child. The programs have only become more demanding and strenuous over time, so I imagine it is even harder now. There are a few programs that are committed to the success of their students, like Princeton I believe. I would highly suggest talking to students at these various programs, finding out which ones do not eliminate parts of their cohorts and where the environment is supportive, and maybe apply to those. If you get in, great. If not, maybe consider doing an MA in public policy, public health, or statistics.
If you do decide to pursue econ, talk to as many grad students in econ as you can to get a real sense of what it is like. You really have to know what you are getting into and that you are willing to take it on. Again, you can do it if you are willing to make the sacrifices, which will probably be time with your child. You will need child care, someone to make sure everyone eats, etc. In econ grad school, you will have to be in class most of the time in the day and studying as much as you possibly can. So if you had someone near by to help that would make it more possible.
Okay, long ramble. Good luck!!