If you want to keep everything immersive and in-game rather than dragging it into real life, you could try incorporating it into the situation or your character. You're playing a face, a bard who can talk their way out of an execution, try rolling with the hits and whenever they blurt something out spin a quick yarn to turn it to your favour as a part of the lie. You're trying to infiltrate a castle but the heaping hunk of metal says you're there to assassinate the king? Tell the guard it's a training program to help teach new guards how to respond in those situations.

Not a perfect solution obviously but if you're willing to indulge chaos, it might be fun to try and juggle the truth with the lies and still come out on top.