I also could have written your post. I did have pretty bad PND but I am not sure whether I would still have felt the same way without it. If you read the book 'one sock, two ?? and no hairbrush' )or something like that the author raises this as a common feeling when having a second child.
With me it was when DD was born. DS was so loud, big, energetic...I just wanted to sit, cuddle and feed newborn DD and it was like he was impinging on this. I wanted him to go away. Of course he was adjusting too with some pretty awful but normal behaviour whilst DD was so simple and easy. I resented the fact that he kept me awake, took up my energy and my thought, stopped me sleeping etc.
It did pass as I got better but even now sometimes I just crave time with DD. I think this is because when DS is around she doesnt really get a look in. It took me a while to feel ok with it but I actually kept DS in nursery part time (I was going back to work) where he burnt up energy and I got my cuddles and to act like a first time mum and the positives that had.
I think sleep deprivation plays a huge role in this. The first time around you kind of get to catch up on some rest but with a toddler and an energetic one at that you just never get that time off. She is playing for your attention but in a way which is exhausting you further and further. You realise this but it doesnt make it any easier.
What my HV said to me always stuck - if you were a bad mum and dealing with this wrong you wouldnt be worrying about it. By posting here, by crying, by worrying at home you are showing what a kind caring mother you are. Being a mum to two is hard. When one or more of those children start behaving in a challenging way or if you have far too little sleep it can be incredibly hard.
I would really suggest you talk to your health visitor. Is she nice? Can you talk to her? Is there anyone who can give you a break from DD? Does she go to nursery at all?
I really doubt you are falling out of love with her...you are not enjoying being with her perhaps because you are exhausted and her behaviour is very trying...but you know you still love her deep down. You wouldnt be posting this message if you were not.
So what if you would be happy for someone to take her away for a few days? Thats hardly a sin is it! Its not like you want someone to take away a newborn baby or you want her gone for months...you just want a break. I bet in reality you would miss her as soon as she went (weird thing motherhood lol). As I said - nursery was my sanity. I knew DS was happy and playing, I got my time to myself with DD. Could she go to nursery? Could a grandparent or someone else even have her over night on the weekend?
Her behaviour...what you are describing in so so normal. My DS was like that before DD was here...normal toddler...but didnt mean he was unhappy. I too worried it was because he was deeply unhappy...but she is nearly 3 - I think its in their instructions to behave like this and although it may be exacerbated by the new baby I bet she would have acted in a similar way without it.
Also DS is only 10 weeks old. From my experience and friends experience everything seems a bit up in the air at that stage. By 3 months...4 months..maybe 5 months everything seems to be so much easier. Although it seems like they will never accept the baby they do. DS loves DD now. He has stopped the insane tantrums (well reduced them!), stopped biting his friends at nursery (!), talks happily about his sister instead of saying he hasnt got one (!!).
It will get easier but if you think you need support, ask for it. I hope you are feeling better and getting more sleep xxx