I find it curious that you find it so important and bothersome that he said 3am if it was 2am. It makes zero difference. Staying that late arguing is a bit strange and I’d feel it was all night too. Perhaps he exaggerated. I don’t see any benefit in texting fight especially late at night. There are more serious issues in your home than him being wrong on the time

Your partner is incompetent parent. He can’t be trusted alone with kids. He is unable or unwilling to be responsible for them. You said he can’t even baby sit them when you are out of the house for 2 hours. I’d not allow him alone with kids