This is really interesting, I expected to see people saying 'it's a phase, ignore and it will pass, don't encourage fussiness' etc but it appears the world has changed, in a good way (or all the veggies are posting here and few others).
I think your recent update is a good way forward. Having a few sausages and burgers of both sorts in the freezer might be good, as they are so easy to throw in the oven, if either interest group feels something is lacking.
I was aware of vegetarianism as a child (in fact I recall with shock my mother feeding sausages to a friend whose family were veggie, saying 'you don't get much meat at home do you?' when we were about 7. I hope the friend wasn't exclusively veggie!). I was also aware that we ate what we ate and it wouldn't have been an option.
At 11 I declared that I wanted to be veggie and my mother refused point blank. She believed it wasn't nutritionally adequate (she used to make comments about an adult friend, brought up veggie, being 'small and weedy') and that while she was responsible for my nutrition I would be eating meat. As a working mother, she didn't want to spend extra time cooking either. I held out for about a week before giving in to chicken noodle soup(!) but it was clearly not workable.
At 14 one of Mum's colleagues, to whom she'd obviously talked about my inconvenient interest, gave her a veggie cookbook for Christmas. That was it, I started cooking and have been veggie ever since. The sad thing is that because she remained resistant, seeing it as a 'fad' for about a year, which I found insulting of my reasoning capacity and a sign that she didn't know me very well so which reduced my respect for her, and she still wanted to eat her food, we just cooked separately from that point on, which was massively unsociable and detrimental to our relationship.
Despite all that and knowing that six-year-olds can have an instinct for things that they would otherwise arrive at later and will always gravitate towards, I wouldn't assume that your DD definitely means this - children are influenced by friends, other people and passing enthusiasms. I would offer some options and keep talking with her about it. Information is power and knowing what meat is, where it comes from and that there are different methods of production will allow her to develop and even change her view, for good reasons, as she goes along.