She clearly is able to feed herself if she is doing it when she likes the stuff.
Your issue is more to get to eat what she isn't keen on.
Unless she has some issues going on (dc2 was constipated and ate very little for a long time), then she is basically a very fussy eater.
I find it interesting that she doesn't fuss as much when your DH isn't there. By any chance, is it the samw thing when you are out and about or someone else is there and all the attention isn't on her anymore?
I'm wondering if it wouldn't be a good idea for her to eat on her own earlier and then she can just join you when you are eating and having little nibbles with you.
Another way that works well is for her to eat with other children, all of them together. They always react in a very different way and aren't as fussy ime.
YY to make it as low key as possible and trying to identify the trigger. PP have talked about not recognising hunger, lack of fine motor skills etc... so I would make that your first angle of attack. If you know the trigger (eg your DH pandering to her or you getting into an argument about how to deal with it) then remove the trigger.
Don't give in only giving her the foods she eats. But you can give her that and insist she has at least 3 mouthfulls of the rest. Then leave it at that. Carry on introducing foods she isn't keen on mixed with the others. She wil get used to them.
I do get what you are talking about. When dc2 was constipated, the only thing he was happy to eat was cucumber. Not great for a 2~3 yo trying to grow.... (and yes he did 'loose' a lot of weight he didn't put any for more than 1.5 year moving from the 90th centile to the 20th centile...). It's worrying and you want them to be better especially when you can see they are tired from the lack of food.
The balance is hard to find but the mix of somethng you like with something you aren't as keen has been what worked for us.
It wasn't quick though....