I don't really see that it is that important to let the kid know you are in a romantic relationship with his mom. You don't live together and so he really should not be seeing more then a close friendship between you and his mom.
I think the lying situation needs to be handled between mom and the ex. She needs to let him know that she does not appreciate his attitude when the kid mentions you. She needs to let him know that the kid feels he can not be honest that is not good. The ex should not be laying those kind of feelings on the kid anyway cause probably does not understand anything except his dad getting mad at him.
good luck