Irregardless of the Police comment. I'm commenting more on the ffact that you don;t need a degree. You don't need a degree to do many jobs, vocational jobs, it shouldn't be an insult that you don;t need a degree to do a job, if anything it plays into the private education sector who are very happy to have academic inflation as it means they can charge more for degrees.
Jobs that didn;t need a bachelorrs degree are starting to demand one, and jobs that did require bachelors degrees now have preference for masters degrees, even though the vast majority of what they will learn will not be applied in their work.
It's a bad trap to fall into to use that as an insult, it's one of the prime reasons US university tuition fees increases are one of the highest inflation busting price rises in the past 20 years.
Not if it involved mocking other people who don't have degrees (plumblers, electricians, car mechanics, sales agentsm cleaners, clerical staff, catering industry etc.), you;re effectively also saying anyone who doesn;t have a degree is only dumb, or the reverse insult anyone who doesn;t have a degree is only good enough to work for Police.
I you want to look at it at this angle you surely can but I would hope that most people would understand it is a relative comparison to other civil servants who tend to be top of their respective classes. If such taunts were made at plumbers, fitters, cleaners road layers etc you may have a point but the targeted nature of the taunt clearer identifies it as one particular group of well paid, underworked (most of the time) civil servants.
But again if you want it to be a general class struggle type of thing you can.