I agree with the other posters about how it sounds like a similar situation to the other member recently (and also similar to a couple of other members over the years whose relationships have not worked out). These could be warning signs of what could be ahead for you and perhaps it's an indication that maybe the relationship isn't meant to be.
Just a few thoughts from another point of view though:
How is your fiance feeling about her visa denials and about moving to the US (or you moving to the UK)?
I'm just wondering how she has been affected by all of this. Perhaps she is projecting her frustration about the situation onto you - her annoyance and impatience with you maybe because she feels like she is the one doing all the hard work, not you.
After all, she is the one who was about to uproot herself and her daughter from a good life in the UK to live with you in the US. She was going to leave her NHS career and her life in the UK, not to mention taking her daughter out of school and away from her friends and her life, to become a student in the US, completely reliant on you for funding and accommodation.
She has suffered through two US visa denials which has now affected her immigration record and is preventing her from returning to the US again for a number of years... and if she remains in a relationship with you, she will be affected by that for the rest of her life and may have problems entering the US from now on (at the very least, she will need a visa to visit the US for the rest of the life). I have applied for 4 US visas in the past (and will require a visa to visit the US forever due to not being eligible for the VWP) - 2 student visas and 2 visitor visas - and the process is nervewracking enough without factoring in any visa denials.
She may feel that it's unfair - that she went through all of that because you are unwilling to give up your job, your home, your pension and your great life in the US for her . As you have probably now realised, it can actually be much harder for a UK citizen to move to the US than a US citizen to move to the UK, especially as a same-sex couple.... and essentially, you are making her do all the work and sacrifice her life in the UK because you don't want to move.
You've said before that moving scares you and that change scares you, but what about her? What about her daughter? They were going to leave the only life they'd ever known to move to a new country where they knew no one but you. Did your fiance actually want to live in the US (and leave her other child(ren) behind)? Did she really want to study in the US or was she only doing it because she couldn't qualify for a spousal visa and you were unwilling to take the much easier route of moving to the UK on a civil partner visa?
You have mentioned that you're going to look into a civil partner visa for the UK, but it sounds like it's something you are doing because you have to and not because you want to... it doesn't seem like something you are happy or excited about - and going into it with resentment or unwillingness isn't going to help the situation and may put even more strain on the relationship (i.e. how is it making her feel that you don't want to move to her country, that you want her to sacrifice her life for you, but you aren't willing to do the same for her? Could that be why she is acting the way she is ...and is it only going to get worse with time?).