Jen,

Then I would do what the rescue organization says. A happy mama is a healthy mama. Just don't be surprized if she moves her babies around. It's normal. Make her a nice birthing box lined with plastic and towels or blankets. When you cut out the opening, make sure the box is deep enough to keep the kittens IN the box, and low enough so mama can take a break now and then.

As far as handling the babies before 6 weeks, I don't see a problem as long as the mama is okay with it. The more handling the better socialized they'll be. It'll also gain the mama's trust.

You're doing a great thing, Jen. I only wish I could do the same. But with 7 cats it's a little difficult to close off a room. Keep us posted on their progress!