After one lie, they start fact-checking your true stories too.

A lie does not stay contained to the subject it was about. Once someone knows you are capable of it, every previous statement gets reassessed and every future one arrives with a small delay while they check, which is exhausting for both of you. That is the real cost, and it is larger than whatever you avoided.

If it has already happened, correct it yourself before it is discovered, which changes the story entirely. Then accept a period of being checked without complaining about it. Consistency over a long stretch is the only thing that removes the delay.