I really loved Kyle's enthusiasm at the fundraiser, she was excited and happy and kept hugging and hugging Yolanda as the money was being raised. It seemed like Yolanda, Lisa and Kyle were hosting or something, they seemed to be going around getting people's donations.
I think the primary reason Kim was irritated with Kyle because she expected to her to join the pile on with Lisa R, which is why she asked her why she was hiding behind her jacket. She was more or less saying "aren't you going to say anything?". And as the trip went on and Kyle spent time with Lisa e.g. riding in the limo with her after the plane ride from hell, and leaving with her to the fundraiser, and appeared to actually be enjoying herself, it irked Kim even more, ergo the "lack of support". Because of course, Kyle should fall right in line and do what Kim wants at all times, that is the support Kim is looking for which she is getting from Brandi and loving it.