BACKGROUND
I have been dating my girlfriend for a year now.
She left her house 5 months ago.
She has a 4 year old son
She lived with her ex-partner 10 years
The boy knows me since the beginning of the relationship
My relationship with the kid is very nice and we are close friends
We have not told the kid that we are in a romantic relationship
The father's kid hates me and when the kid speaks of me he gets very angry so the kid started telling the truth to avoid conflict between his father and me.
We are worried that the situation has led the kid to use lies in order to avoid conflict.
We have had difficult moments with my girlfriend and once she went back to her old house.
We do not live together.
She lives with her family.
The kid spends the weekends with his father
Last week we went on a trip with her family and the child, my girlfriend and I, slept in the same bed.

QUESTION
How should we behave in front of the child?
When would be the right moment to tell the kid that we are in a romantic relationship?
How to teach the kid that he does not have to lie to solve the problems?