I've got one of those, op. She's 8.5 now and gets upset a lot more easily and a lot more often than her siblings. She's a real home body, and likes things to stay the same. She hates being away from me overnight, for example, whereas the other 2 love it. She can't help but burst into very dramatic tears when upset.
She will also regularly tell me about times she felt really sad, nervous or scared in the past - she might not have had the words or maturity at the time to do anything except cry, but she remembers very vividly.
She can have episodes like you describe where she just keeps getting upset about something over and over. I don't see it as manipulation in any way. She genuinely finds feeling big feelings difficult.
I will always listen and help her deal with the feelings when she's upset about something, but I set limits. So after we've talked about it and faced up to the fact that whatever it is has to be the way it is and had our hugs etc, I've had to get very firm and close down any further upset and discussion. "I know you are upset/you don't like it. It's hard for you. We've talked about it and I know you understand why it needs to happen like this. We cannot talk about it anymore. I'm going to go do xyz now." I might suggest she draws a picture or writes something about what's going on in her head.
It's been hard, but she's getting better at parking the thing and accepting the rational reasoning and moving on. It has also helped a lot to talk about feelings in general (when she's not upset) and reassure her that it's normal to feel apprehensive, worried, anxious, etc. Everyone does. And talk about the ways to deal with it.
I think your DH is right, to an extent, in that you have to help her set a limit and find a way to move on. But I don't think it's fair to say its manipulation. It's tough to deal with, and you have my sympathy.
One last thing to mention is that my older dd goes completely silent when upset or angry. She cannot describe her feelings, which worries me more in the longer term. So, while she is less dramatic, I don't think it's necessarily better.