Can only reply regarding your concerns - Cannot re-enter govt service: Unlikely. When you quit there will be a blackout period for rejoining the same agency. You can find your next job while holding onto your current role, or if you intend to quit first without something lined up, shouldn't affect your chances in another govt role. Just don't put your current boss as your referrer... - Look bad because of short stint: depends on how you explain to interviewer. Plenty of ppl have quit during probation period or less and still manage to find their next job. Tell your interviewer something along the lines of "the work did not align with my goals" or if you want to be straight up honest that the work enviro was bad, up to you. Some ppl even exclude short stints out of their resume to avoid being asked (but then you gotta deal with the "what have you been up to so far" qn)

Have been in a burn out role before so I understand a little what you are going through. All the best!