I would go for option 5:
Check craigslist for available EEs. You would be surprised at what you can find even this time on year there.
You couldgive up your two remaining roosters, keep the leghorn and gold lace seabright hens and add two EEs.
Good luck and don't give up.
What are the dimensions of your coop? For 6 chickens, you would need about 20 sq. ft. of floor space.
The general rule of thumb is at least 4 sq. ft. of floor space in the coop per chicken. For example, I have a 6 x 8 ft. coop with 11 chickens. I've also heard that it should be 10 sq. ft...
I'm sorry for.your little girl and for your impending loss. :(
Call your local humane society/animal shelter. They will probably put your rat down for free.
Congratulations on the egg!
Looking at your pic though, I see chicken wire in the background. I don't know predators are in your area, but chicken wire is only good for keeping chickens in, not for keeping predators out. Hardware cloth is a better option for keeping predators out.