My teen has ADHD and ASD and 3 was the hardest age. No usual strategies worked with DD - did not give a shit about consequences. DD also hated me showing an attention to friends and cousins - so I imagine having a younger sibling in the house would have been very hard.
What helped then (and continues to help is)
Building communication skills (speaking and 'reading people') - it is very frustrating to be 3 and not be able to express yourselves, even if NT. We used a lot of Makaton alongside verbal speech. We also realised that DD generally assumed that she would be told off every time I said her name and simply cannot read faces or cues to assess a person's emotions. So even now there is a lot of explaining my tone - Mum is tired, Mum is cross, Mum is not angry with you etc. The more they can recognise and verbalise their emotions, the better, but this is something DD had therapy to help with when older.
Exercise - keep them busy, and try activities which offer the right sensory input for your child. DD needed a trampoline and from the ages of 3 to about 12 would spend hours bouncing. Other kids like to spin or swing, but it does help them regulate a bit.
Strategies - if you know that the usual ones don't work, forget them. Just find ones that do - distraction, bribery (professionals told me to use this to change a habit that was stuck and said its fine as a parenting approach, as long as its not your main one!). DD would never back down, and the more I went head to head the worse it got. The advice that stuck with me was from The Art of War - do not back your enemy into a corner, they will go down fighting - give them an escape route!! Ultimatums didn't work - an exit route always did.
And finally - the decision to pursue a diagnosis is up to you. I agree, it can be hard and scary to open yourself up to professional scrutiny. The other day it occured to me that now DD is nearing adulthood I don't need to be worried that someone will decide I am a terrible parent and whisk them off.
We got a diagnosis aged 7 - I said to the school that DD probably looks fine, but in two months I am going to ask you about ASD. It worked well as they didn't give a kneejerk reaction and when I did ask they had a lot of examples as to why it would be sensible. Its ok to do in your time, but with waiting lists, assume once you start the process, it will be years before anything happens.
Don't underestimate the power of doing something silly to break yourself out of a rut. Eat pudding for breakfast, wear silly clothes, give yourself a break from the day to day.
Oh and if they're hungry they're horrible. I used to meet DD at the school gates with food and water (wouldn't eat at school) - it was pointless even trying to speak or do anything until she'd eaten.
It does get easier.