We just don't use "not to" that way. We use "to" to show that someone is doing something for a purpose ("I called him TO tell him I was taking a cab"), but we don't use "not to" in the same way. You can say "I did took a taxi in order not to be late," or "I did took a taxi to avoid being late," or "I took a taxi so that I wouldn't be late." But you can't simply say "not to be late."

Gray gave a good answer. I’m just going to add that another possibility is the so-called split-infinitive: “I took a taxi to not be late to the meeting.” There is a grammar myth that the split-infinitive (which, by the way, goes back to Old English) is incorrect, but in real life, educated native speakers use it very frequently in conversation.