'DH wants to make it work, not the least cos of what friends and family will think of him'
'But my family/friends/community is a closely knit one and I don't think I can bear a divorce when I've only been married less than a year.'
You both seem very concerned about what other people will think if your marriage breaks down. Surely the important issue is how you feel about each other and whether you can trust him?
His ex is the mother of his children and will always be part of his life. The fact that the police warned you regarding your communication with her, and that your dh took her side, should tell you that you were seriously overstepping the mark with your comments to her. No, he shouldn't be lying to you about time spent with them, but given your history of vitriolic reaction, him covering things up to keep the peace is somewhat understandable.
Your marriage doesn't need to be doomed. Doomed is a word that suggests it has no chance. It does have a chance, but marriages need effort, compromise, mutual understanding and respect.
Both of you need to agree to turn over a new leaf and be totally honest about everything, and you should agree to behave in a more dignified manner towards the mother of his children, while putting more trust in him. I do think he has made mistakes here, but then so have you.