Just wanted to say I am in a similar situation, however not quite as severe as yours due to a large age gap between mine, so the oldest is able to understand and express himself.
I'm mentally exhausted having 2 children even with a 9.5year age gap, mostly due to DS2 still being a terrible sleeper, working and limited childcare.
If you're struggling to get 1:1 time I would say get someone to take baby out for a walk and leave you and toddler together or get your partner to take baby out for a walk at the weekend, try not to give any warning so 3 year old isn't trying to join the walk. Obviously he may have a tantrum the 1st few times but it should get easier.
Or if you have the opportunity take him to the park or for a walk. Maybe try a new thing that only the two of you do- go and feed the ducks followed by a hot chocolate/babychinno
Take his lead with play or books. If he ignores you then start playing nearby quietly or read aloud to yourself as it may catch his interest. Do bathtime ir something fun.
He's obviously not adjusted well to the new addition but it's so important to try and get that bond back.
Surprise he each day with a big cuddle and kiss when he's not expecting it. Sometimes a bit of rough and tumble on a bed can be quite fun, let him crawl then pull him back by his legs and tickle etc
Lots of small things make a big difference.
Obviously the baby will get more attention, do you think you give the baby more kisses/cuddles than your 3yr old? do you actively give 3yr old affection without him seeming to ask for it? Does he get involved with the baby much?
If you're exhausted or mentally drained, you might not be giving your 3 year old enough affection/attention. It's almost as if his reaction of ignoring you is his way of protecting himself from being rejected/not having his emotional needs met. I don't mean this to offend, just trying to think of a way to explain his behaviour.