When I was young(er) I used to do stupid stuff like described above, also. It comes with the putting young people on fast motorcycles.
The original question, about what to do about the reckless driving arrest. All you can do is show up in court and lie. I mean, you cannot show up in court and say, yes, I was getting it big time and the officer heard right. You have to say, no, I accelerated fairly hard and the officer heard the pipes, but I never exceeded the speed limit or operated in an unsafe manner. He can make a point he had to do 125ish to catch up, but without a distance measurement with which to compute your speed, it isn't sufficient.
Then you get to find out what the officer has factually. Which may be nothing. Or may be something.
And no, an attorney won't help, thats a waste of money for something which will turn out to be either the officer has nothing but what he heard (dismissal, or maybe a loud pipes ticket at worst), or he does has some evidence, in which case you lose.
If the officer states that the distance he covered is X miles, and gives a computation of your necessary speed to take him Y minutes to catch up to you at 125mph (easy math) you will be found guilty.
Living the Gypsy Life