If you were still married to this man and he came to you and said he wanted a divorce to be with someone he loved more than you, would you have gladly shelled out $3,000 and said "Sure! Let me pay for all of the legal fees to make YOU happy!" I don't think so.
I've been remarried a year and my husband's ex-wife was overpaid $7,000+ in child support money and is still being a witch. It took us a year to get the youngest daughter who is now 20, married and pregnant off my husband's garnished paychecks and we just saw that money coming back to us this March. Ex-wife did every trick in the book so she would keep getting that $300 a month and that sure didn't stop the daughter from expecting us to help her financially and we told her to go to her mom who was still getting child support on her even though she was GROWN and chose to be a high school dropout with a loser husband.
Your kids are small now and maybe you don't see the big picture. So the ex and his new wife are having a hard time financially. That was their choice to end your marriage to him and take vows with each other. Too bad they get the "poorer" end of richer or poorer but you are not legally bound to give them anything outside what is stated in your divorce decree.
If your ex were to lose his job, that $3,000 would've helped your current family make bills, etc. and now it's GONE. Keeping the peace doesn't mean you have to be a doormat. You can be polite without hurting your wallet. Think of the example you set for your children. Would you rather them see you get bullied by their dad or applaud you for standing up to him? If your daughter were in the same situation would you tell her to "pay up" too and allow some man she thought she'd spend the rest of her life with to manipulate her, cause her unneeded stress, etc. if her husband left her for another woman? Doubt it. Reinforcing the guilt trips, anger, bowing down, etc. is only sending a message to your children that "It's OK" to be treated like this and play the "victim" not the "victor."
I have 6 stepkids from 2 different ex-wives to deal with and trust me when I say you may not think the children remember much from their childhoods and what divorce did to them but they do. Mine range from age 31 to age 17 and there's a lot of bitterness because of the broken family unit to the point when holidays come it's all about choosing sides and usually we don't see them as the mother is the preferred parent. Time passing doesn't mean that everyone is going to get along and be mature either. If you think it's bad now, wait til your kids play you and the ex-husband against one another to get what they want. It's not pretty.
Take up any grievances the ex has on finances with a lawyer and document all emails, phone calls, text messages, etc. I hope things look up for you soon.