Firstly your children have to be told the truth.
How else are they supposed to navigate this?
Oh yes, that's right... YOU end up being the bad guy.. Keeping daddy away.
Your children have to learn that when someone lies like that to the family that it can't just get ignored and that at the very least you have the power to choose who you have living in your home. You get to choose because he chose to lie to you all.
Look at Airbnb, he can rent an apt or a house for the duration of his stay.
Do not let him back in your house, it will kill you. (Or you'll kill him...)
I had my ex here on and off this summer for the first time in 4.5 years. (He was abusive and a waste of space, not a cheating low life liar)
I WILL NEVER EVER LET THE BLOKE INTO MY HOME AGAIN.
For reasons of his behaviour on the second to last day, I will also never ever let him care for our son.
Your ex has to pay the price, the children have to understand why you are being so firm about this.
You have not given them enough information for them to be able to process this or understand your anger.
You will be made the bad guy, trust me. Put a stop to it today by telling your children the age appropriate truth. Tell them separately if need be so that each had the opportunity to ask questions and have the truth said.
If I could tell my 5 yo that his daddy had to leave because of abuse, you can tell your children that their dad broke his promises to you and that as a result to can't be together.
Before I got to the end of my sentence, My son actually said to me "is it because dads shouts at mummy"
Over the following 5 years I have answered his questions more he appropriately. Now he has seen his father this year, he's under no illusions.
Your ex did this, remember. Don't take any blame for any of it.