Okay, so it's impossible to make any generalizations on what its like to article, or to work, for "government". There are different levels of government, different jurisdictions, different practice areas. Even within the same jurisdiction, different individual managers can make quite the difference.
So all I can say is that the risk is not zero.
One thing you can say about governments is that they are sticklers for following the law. If you show up with a doctor's letter and documentation saying you need to work one day per week from home, I'm willing to bet they'll accommodate. What I can't say for sure is whether or not it would impact your hireback.
Just within my own agency I've seen government be super supportive of people going through personal trauma, going on lengthy leaves, and welcomed back with open arms upon their return and with no harm done to their career. I've also heard of personal accommodation requests being refused (all second and third hand stuff). And I have seen lawyers who, after returning from a medical leave and being cleared to work, being let go on a without cause basis.
I think it sometimes comes down to how valid your underlying reason is perceived, as unfair as that may sound.
If you have a good relationship with your principal I would raise this question with him/her. Do not make it a formal request, but in private mention that you were thinking of making this request. Be open about exactly the fears you mentioned here.
I understand why you don't want to mention any details about your medical situation on this forum, but a lot of government offices are fairly gossipy places. Even if kept in secret why you're getting this special accommodation, the reason might leak out. Even if it doesn't leak out, there may be a lot of gossip and speculation about why you get this accommodation. The better approach might be to be open and upfront with everyone about the reason, but you may have reasons why you don't want to do that also.