An interesting exchange - and one which doesn't actually address the issue at hand. Should flying any flag deemed offensive by another person lead directly to dismissal from public service. I suspect it's there in her contract about bringing public office into disrepute etc - but as someone who is an old fashioned liberal I find it a little troubling. That goes for all flags. If you were working in a church and you flew a trans-inclusion pinky-blue flag and the church fired you, would that be a different matter? It wouldn't be a different legal matter, they're identical and opposite in nature - but the average activist may consider them to be different matters.
To some extent this seems like 'whataboutism' - but as someone who is neither trans nor a US Citizen, I don't really have a horse in this race but I don't like the legal vagaries you've alluded to - it allows the legal weaponisation of offence and it allows that offence to be universally applied. If you take the bipartisan nature of the subject out of this matter and simply explore whether political support (of any kind, flag flying or otherwise) should be directly related to job capacity, the issue is quite interesting.
Ask the majority of 'liberals' whether police officers should be allowed to fly Confederate flags and they will tell you 'hell no - because it's offensive', ask them whether church youth workers should be allowed to fly trans inclusive flags and they'll tell you 'hell yeah, because it's empowering'. This failure to see that bias about the process negates fairness about the process is part of the issue with communication between left and right. Either everyone can fly flags or nobody can - and society needs to pick, rather than 'pick and choose'.