david - have you contacted BG probate dept about the bill, as that sounds wrong.
My DM died last Feb, DB contacted them a week later. Account frozen, I asked him if he needed to tell them we had probate (end of April) but since house was then under offer, he didn't bother until completion day (end of June). We heard nothing in the interim and then got a final bill after completion.
And even better, they wrote to him with a refund in January, as apparently they didn't increase her DD in a timely fashion when they should have done, presumably last winter when she had heating on permanently. Was a nice little bonus long after completion.
They shouldn't be asking you for any payment until probate is granted at the very earliest.
Cant find paperwork, but we had oath info back fairly quickly, within two weeks, was able to sign a week later, and Probate grants recieved about two weeks after that. All very quick.