I am a granny and a teacher , and been involved with small children in lots of ways over the years. So firstly, at that age she is not able to express exactly how she feels or ask relevant questions about things that bother her. If her parents havent explained the chair use to her in a way she can understand she may have worries about it , as it is something she is not used to seeing. If you are stressed now by the way she behaves your body language may be giving her mixed messages. As for the shush , there are a couple of things that may be causing this. She may never have the opportunity to say this to others and now she has the chance to say this and get a reaction, which makes her feel in control. Alternatively she could be just being rude and meaning to be like this. At two if you remember your own children they are just finding out the great ability to say NO and take some control of their own lives, which up till then then have just had to do what adults tell them to.
Ideas that might work to improve matters.
If you can have a couple of books suitable for a child that age with you. So the paperback version of The Very Hungry Caterpillar, would not be hard to hold and you could turn the pages and read it aloud but in a quietish voice. Ignore her "shush"ing and carry on turning the pages and looking at the picture. Dont appeal to her to join you, or ask her if she likes it, but let her be drawn to the story and want to look at the book. If she comes close to you, dont comment on this but just show her the pictures and respond to what she says.
Have you got a light tray that would be comfortable for you to have across your chair? Dont know if you have a Button Bag with all your spare buttons in . We used to love spilling the buttons out and making flowers with them and patterns.
So anything safe that can be sorted and made into pictures or patterns. Perhaps her parents have a couple of very simple jigsaws that you could make up, or you could make a very simple jigsaw from paper , by looking for a brightly coloured picture in a magazine, cutting it out and into large pieces. Do the same with another picture and then mix the pieces together and put them mixed up on your tray. You then start looking for the pieces to make up one picture. Just carry on and dont expect her to join you and she may come to look of her own accord, or you can deliberately drop one piece on the floor and ask her to get it for you. So just be yourself, dont pressure her to join you in anything but be welcoming.
Again with the tray you could have playdoh or I have made pastry in the past and had different shaped cutters . So long as your tray or board is clean you can cut out shapes or have a little baking tray to cut out pastry and make jam tarts to be baked and eaten. We are all basically animals and if you look in the animal kingdom, anything new has to tested out slowly and you have to wait for the animal to approach you in its own time when it feels safe and comfortable to do so. Even if you have food for an animal its instinct is to take things slowly and make sure it is safe to approach. If you think of her in that way you will understand that she is not showing that she doesnt like you. It is much more likely that she doesnt see you often enough yet to have become accustomed to finding the chair immediately approachable, and at 2 if she doesnt see you every day it still can seem strange to her. So just be yourself and do what you normally do and let her get used to how and what you do and let her set the pace. I think you will find as she comes to connect your chair with fun things to do and lovely stories read she will relax and you will get along fine.
Dont forget that once she is used to the chair you will actually be more approachable than a standing adult. Also do you have any other children who visit you and who are used to your chair and behave normally and bring things to you? If you do then perhaps she might visit at the same time and she will see there is nothing to worry about , and of course she wont want to be left out. Do you play what I think we called Kims Game with other children? Where you have objects on a tray and then remove one and they have to look to see what has been moved? That would be a good game to play with her when she is more used to you.
My only word of warning is I would never try to coax her with things like sweets cakes etc. It would be a bad precedent to set and you want her to choose to be with you not what you have. Good luck and I would love to know if any of this works for you