So sorry you (and your pup) had to go through this, OP. 37 degrees and they were worried about your dog getting too hot? Have you ever sat in a car at that temp? No way you could get too hot unless you were strategically placed so that the sun were shining right on your head and you had no way to move. Even then.... I just don't see it happening.
One of the many reasons that I love winter is that I can take my dog more places. I am extra, extra careful about taking him anywhere if it is over 60 degrees (F) out so summer trips are hard. But, in the winter, he sometimes comes with me for a grocery run just cause he loves car rides. He has a blast looking out the window on the way there, naps while I'm in the grocery, and has a blast on the drive home. I can't imagine how I'd feel if I came back and someone had removed him from the car for his own "safety". I hope the charges against you are dropped!