All they need to do is say, "Oh, no unanimous consent. Let's just vote on it..." and go down the list of senators. IF it's taking a bit longer, it's the fault of the senate.
That's not exactly right.
The requst was for unanimous consent to pass the bill without a full hearing and debate. Without Bunning's consent, the bill would require a full hearing and debate on the senate floor. When Bunning objected, they COULDN'T just say "ok, well, then lets take a vote, right now, you curmudgeonly old fucker." Senate rules would then dictate that the measure be scheduled, debated, and then voted on, which could have taken weeks.
I don't see the timing as Bunning's fault. You were right,they certainly could have scheduled the bill for full debate on the floor, earlier. I imagine they didn't think they had to, though, and so they got caught out by the short hairs on a procedural point.