When you say:
I offer you the option to not travel
the “to” after an option is a preposition or is it the “to particle”. We know that we negate the infinitive by putting “not” before the “to particle”.
In this case the “to” is the “particle.”
This is because the correct grammatical construction would be: “I offer you the option not to travel.” Not always comes before the to particle, and whenever to is in front of a verb it is a particle and not a preposition.