I wouldn't think twice about it. WE used to have an embarrassing 20k in savings. Over the years, I have convinced my wife we didn't need nearly that much, and since we never paid PMI in the first place (20% down), we invested that money and are now down to 6k in savings. Because we are frugal, this savings number grows by itself anyway, and we have to periodically invest the surplus to keep it down.
Do a financial analysis, the rate of return on not having the PMI is far higher than your savings account earns, if anything. It's likely a pittance. Pay off the extra ASAP, before you log off your computer, then focus on replenishing the emergency fund with the savings and future frugality.
Like you said, you work for the government, so your chance of getting fired is about 0 to negative %.