As a barrister, I worked closely with the TSols/GLS/GLD for over two decades (circa 1990 to circa 2013). My observation of Government lawyers was that they obtained hands-on experience of big issue cases when at a relatively junior level, whereas in big law firms they would still be carrying a bag at the same level of seniority.
Some Government legal work is mundane and tedious, but some of it involves major issues of policy, and acting for and reporting to senior civil servants, Junior Ministers, Secretaries of State, and sometimes even Number Ten itself.
Towards the end of my period as Treasury Counsel, I had security clearance and got to do Secret Squirrel stuff with Five and sometimes even Six, and worked with some quite junior GLD lawyers who had the same security clearance as me. I could tell you about that but then I'd have to get Six to kill you (and I'd go to jail, but not for murder).
There were and no doubt still are big issues with under resourcing, and the pay is not what it should be, but as noted above the work/life balance beats that in the big law firms, and the quality of the work can be very stimulating.
As for showing ability in public law, you would probably need to demonstrate experience in JRs, or advisory work on public law in its wide sense.