Hi there, this is Dr. Elaine.
It sounds like Laura has prolapsed her cloaca, which is the outlet through which urine, feces, and eggs all pass in common.
Hens in lay can experience egg-binding, calcium drain (which can lead to weakness in the muscles of the tract), even severe diarrhea which causes excessive straining.
This is a medical emergency, as the tissue can continue to prolapse further until actual intestines do protrude. Also, the exposed tissue is going to become infected, dried, and none viable the longer it remains out, meaning instead of being able to replace it inside her, it will need to be excised.
Many veterinarians (not necessarily bird specialists) can do this on an emergency basis, but call first so as not to waste time.
Technically, tissue protruding from the cloacal opening is a prolapsed cloaca, unless it is a tumor or tissue from higher up (both less likely).
Yes, the treatment consists of cleaning (flushing with saline, removing all foreign material, keeping the tissue moist while doing this (sometimes with sterile KY jelly), then gently positing the tissue back inside (it has essentially turned inside out), and "anchoring it" there with any number of suture techniques.
Were we to perform this in my area on emergency, it would likely fall into the 300-400 dollar range, more if tissue removal is necessary. This is going to vary simply because of the unknown complexity once the prolapse is evaluated.