sibling rivalry is a complete red herring in my opinion.
These are two random children of 1 and 2.5 who have been thrown into the same house to live together, so the behaviour is really not sibling rivalry as such... just (very understandable) rivalry.
Your 2.5 year old in particular is at a frustrating age developmentally anyway - even if his language is developmentally normal (and DS's was very far from developmentally normal at that age) he won't be able to express how he feels verbally properly.
So he is showing you how he feels (and it seems has been showing the FC's as well!) - he is terrified and unsettled. He has just been taken from the second set of parents that he's known and plonked in a strange environment where the food is different and it smells different and the routine is different. He may have lost "siblings" at FC's who may well have been older than him so he may well have gone from being the baby to being an older brother to some strange dude that you strange people keep insisting are his brother and who gets way more attention (whether he does or not is irrelevant - a one year old almost inevitably gets more attention, nappy changing etc. It probably sucks to him just now and he's desperate to relax and be babied by someone who really knows him. I agree with someone earlier who said you may need to baby him a lot more at 2.5 he really is still a baby.
Just upsets us when he hits us - I get that I really do - most of us have been in your position (with only one in my case so hats off to you!) but please don;t ascribe adult emotions to a two year old. His emotions are very basic and he has no competent way to communicate them - afraid, angry, happy, sad - that's pretty much it and they don't have the cognitive ability to process them in the say way or with the same subtlety that adults do. That is your job as a parent to expand and develop their brains/emotions/cognitive abilities and in my personal opinion, you do that by modelling not by punishing. (In fact there is increasing evidence that reward and punishment does not work particularly well on children who have already learnt that bad things happen regardless of how "good" you are).
I can't stress how much in your position I would (and did) drop time out completely. You don't want to teach him that if he expresses his anger that he is excluded. You want him to be able to express to you how he feels - you just don't want him to hurt anyone (including himself) in the process. You want him to gradually learn to feel safe with you both physically safe, safe form having to move again and safe emotionally that you recognise how he feels and are big enough to put your feelings aside to put his first. So however upset you feel, imagine that he is feeling about 100x worse and as the adult try to suck it up and teach him how to show it without hurting and he will need lots of repetition before he gets it.
So keeping the younger child safe but (in my opinion not giving him any extra attention just keep him on the other side of you for example) say firmly but calmly...
"I can see you feel cross at the moment, but we don't bite/kick/spit/hurt in our family" then distract distract distract. DS spent a lot of time in plapens being separated from other people, even now 8 years on time out is virtually guaranteed to escalate a situation rather than help. His (recent) psychologist explained (in front of me) to his teacher "When he is in time out all he thinks repetitively is I'M BAD AND THE TEACHER HATES ME, I''M BAD AND THE TEACHER HATES ME"
It's very hard to improve their self esteem once they've internalised this - so I would really resist if you can ever using time out. The only exception in my view would be if for some reason you can;t physically keep your younger child safe.
Sorry thats a bit long but it comes from the heart of a parent currently struggling with a 9 year old with rock bottom self esteem and significant executive processing problems. I have no idea if it would have made any difference if someone had said this to me 8 years ago - as I never did use time out (tried twice and it obviously distressed him to a degree which was out of proportion to the "crime") though I have had to resort to "holding" time in when he was going through very violent meltdowns around 3yrs which would have injured himself if I'd left him to it. But I think the idea that feeling hurt by the responses of any child is to ascribe an emotional ability that they just don't have. Toddlers are total narcissists - all of them, damaged or otherwise, they aren't trying to upset you or be defiant or naughty, it's all about them and how they feel and they're letting you know how they feel in no uncertain terms! It doesn't occur to them (nor should it) that you have any reciprocating emotions.
I agree if he is hitting and scratching at naptime/beditme then he's scared. Children of that age struggle with separation and bedtime is separation to them they have no concept when they are left alone that they will wake up and the day will start all over again. I too would sit with him and holds him hand if he'll let you until he falls asleep.