Firstly, fear of being sick is a real phobia. I know someone who had it badly and she would completely freeze if she thought she might be sick.
Secondly, I know that if I've been sick after eating a food it can take a long time before I can eat that food again. I still can't stand even the smell of baked beans after pregnancy with dd1 who's nearly 12yo.
Thirdly: I struggle with eating. But the worst is feeling an expectation of me eating. The higher the expectation, the harder it is to eat. So try not to let your stress show.
What I'd suggest: (but you may have already tried) give her small amounts. Don't give her a whole plateful. And don't do any expectation of finishing. Think of any mouthful as a bonus.
Try things that may be easier to eat-eg. boiled egg and toast. Or even without the toast, if she'll just eat the egg. Or soup in a cup.
If all she will eat is the same thing then stick with that. When she's feeling relaxed about eating you can add small amounts of different things.
Try not having a set meal time. Put out a plate with a small sandwich, a piece of tomato, a few crisps. Say to her that you'd like her eat that over the evening and then leave it. Don't mention it again. If she takes the sandwich and nibbles it while watching TV, that's fine. And if she's nibbled round the sandwich, but not more, then praise her for that.
Don't worry about the tissue. If that makes her feel better then let her have it. if she'd rather a towel over the pillow then let her. Taking away something she gets comfort from won't help.
If she likes them then go for high nutritional items. Banana, avocados, milkshakes.
But if she isn't getting a full diet in then you probably need to see the GP and ask to be referred. Round here they do have a paediatric dietician (I think that's the term) who is available to help children in similar circumstances. I know one whom she helped and he now eats fine.