I go through this every spring.
I would not recommend moving the nest and/or babies.
As already mentioned, they'll be gone in a few weeks.
My dogs will chase the adult rabbits....never have caught any.
They don't seem too interested in the babies and usually leave them alone, aside from a bit of sniffing around them.
I'm more worried about running the lawnmower around them.
I put one of those small resin patio tables over the nest. It's enough to keep the dogs out, but still allows mamabuns to get in to feed the babies.
Mamabuns will also act as a decoy if she's out there when the dogs are.
She takes off around the pool and the dogs go bounding after. The mama is usually so fast that she gets all the way around and then has to wait for the dogs to catch up.
Last time this happened I was laughing so hard I thought I'd have an accident.
The poor dogs get worn out after the third or forth time around, and then just give up.