Welcome to the Forum...even though you probably do not want to be here!!
As you have already noted, your father's case has "red flags" all over it. I am NOT a physician, but I would agree and urge him to get a second opinion at a place that treats lots of bladder cancer before BCG or any other treatment is started. (A delay will not be a major issue in effectiveness.)
It is considered good medical practice to do a second TURB before a final diagnosis is made, especially when there are questions from the first. During the initial TURB, the urologist is sort of "guessing" where to take samples from his visual inspection. After the initial report is in, he has a much better idea of where questionable areas are and can ensure that he gets proper samples from those places, which include muscle tissue.
Unfortunately, Univ of Virginia, while excellent in some areas of urology, is not noted for expertise in bladder cancer. The same is true of Medical College of Virginia in Richmond. Since your father will have to travel a bit anyway, it would be worth the effort to go to Johns Hopkins in Baltimore which has a superb reputation. Other possibilities include Vanderbilt in Nashville, and Memorial Sloan Kettering in NYC.
It is critical that he receive the correct diagnosis and treatment.
As for the urgency and burning, this could very well be due to the TURB. After all, they did cut little pieces of the bladder out!! Also, some types of bladder cancer can cause this. When all the poking and prodding are done, he should be much more comfortable.
Best of luck to you all.
Sara Anne