You say she runs 2:45 now for 400m. That's 11 minute mile pace. Plenty of 8 year olds who don't do crap can go out and run faster. You also say that she plays soccer and basketball and that she's in good shape. Sorry to say, but that's pretty slow for an 8 year old girl in "good shape." I'm not saying this maliciously just realistically - it doesn't sound like your daughter has the talent to run at 7:20 pace for even 400m (which is what a 1:50 is). Or else the soccer/basketball teams don't work her like the one's I've seen.
But, there's no harm trying.
I think it's OK to have her do "training" in moderation. As other parents have pointed out don't push her but if she's into it, it's no different than having her practice violin 30 minutes a day. It's developing one facet of her. As long as it's only seen as one part and not an obsession it's good.
You don't want a kid to do too much anaerobic stuff. But kids can handle quite a bit of aerobic stuff. Her problem seems to be aerobic endurance anyway--even though she can cover the distance. I would have her do some slow runs as long as she can go a few times a week (if she has to walk a few seconds at times it's ok) and maybe once a week do the 400 fairly fast so she can get confidence and practice.
Don't be bummed out if she still doesn't do well. For one, it's only a small part of life. For two, kids change a lot as they grow. Your daughter may change completely by the time she's an adolescent, so just because she doesn't have the talent now doesn't mean she won't later.
Good Luck.