The "late for a vote" routine is why they didn't give him a sobriety test. Article I Section 6 of the US Constitution says "Representatives shall ...be privileged from Arrest during their Attendance at the Session ... and in going to and returning from the same".
While you may volunteer to take a sobriety test, you may not be forced to give one without being arrested in any jurisdiction I’m aware of. Since his claim that he was “late for a vote” prevented him from being arrested, he was able to prevent any testing.
We have the same provision in the Georgia Constitution (it is supposed to prevent arrest by opposing political hacks influencing votes), and recently had a case where a state judge threw that defense out because of what time it was.
I think Kennedy had heard about that and used the "late for a vote" for the reason that (if he actually has read the Constitution instead of just watching it burn at Kennedy Coven Meetings) the "returning from" part might not be as familiar to junior officers and getting it on the record that he was impaired might be embarrassing to the boy.
BTW, doesn't DC have a "under the influence" statute? Here, prescriptions are included under the heading of influence.