Your post hits close to home because I just lost my 17-1/2 year old 'tiel to a tumor (or cyst) in her lower abdomen a few days ago.
I first noticed it over a year ago. I took her to the vet, thinking she was egg bound. He said surgery would most likely kill her due to the tumor's size and her being a bit underweight. Surgery with small birds is always risky; they only have so much blood to lose in their tiny bodies and it's a huge trauma. She had an ultrasound and it freaked her out; she was non-responsive and almost catatonic just from that painless and brief procedure. I brought her home intending to make her last days comfortable. I figured as long as she was eating, and interested in affection, she was enjoying life and wanted to stick around. Well, go figure, the tumor shrunk and stayed shrunk for about a year, her energy returned and she had a happy year.
About a month ago the tumor was back. Again she was uncomfortable, but still showing interest in food and affection. Last Saturday afternoon she suddenly developed sporadic coughing fits and noisy breathing. In spite of her distress, she devoured her millet and craved attention, signs I took to be positive ones. I spent some time with her trying to comfort her, patting her and holding her. The next morning, she was gone. If she had still been suffering that morning, I would considered putting her to sleep, but she had other plans.
My advice? Give her love and affection and watch for signs that she is suffering with the tumor. If she is at more than just a level of "uncomfortable", take her back to the vets to be euthanized. I doubt your pet's mate will die of a broken heart, but they do miss companions they've had and keep looking/calling for them. Try another bird and see how they get along.