Where am I going wrong with my toddler?

Hi ladies... I'm struggling massively with my almost 3 year old, and I need help!

In January, we welcomed our third child into the world. I also have another child who just turned 5. My middle child has struggled with the arrival of the baby, which I expected, and we've done as much as we can to try and give him time and one on one attention - but it's hard! We have zero family support and my husband works long hours, so I run the household alone.

Discipline-wise, I have a three strikes = time out rule, or if they refuse to partake in time out then I take away a favourite toy for a period of time (20-30 mins). My 5 year old responds beautifully to the system, but the almost 3 year old? As soon as I start to issue warnings, he starts acting out in an extreme way (throwing toys, screaming, slamming doors) and when it comes to time out, it's always a battle to get him to stay there and he hits, head butts and bites me. This behaviour has developed since my last trimester of pregnancy but really amplified with the arrival of the baby. He just gets so angry and I try to help him by talking to him and offering an alternative ("are you feeling really angry? Why don't you stamp your feet and stomp out all the anger") but he doesn't respond to it. I am just really at the end of my tether now and feeling so lost. My youngest is 12 weeks old now but still largely dependent and spends most of the day feeding, so between trying to ensure the other two kids get some attention from me and trying to keep them fed, changed, clean etc - it's really hard!!! I stay up late at night cleaning and preparing food when the kids are in bed just so that I can eat, because if I didn't, I'd starve! I am usually pretty calm and in control, but it just feels like everything is slipping through my fingers and even though I have some friends I could talk to, I actually just feel really alone.

I feel like I've failed my toddler and I wish I could help him through this adjustment more. I never went through this in helping my oldest adjust to the arrival of my second child, so I can't tell if maybe my oldest is just a rare breed of Angel child or if my toddler is off the scale in terms of normal behaviour.

What do I do?! Is there hope, or is it a stalemate?!