I don't think your mom's symptoms have to do with her thyroid. Sounds like she may have a lung infection. I think seeing the pulmonologist is her best bet. I would have said these symptoms sound like they are related to her congestive heart failure, but they seem to have ruled that out.
Also, don't let them mess with her meds too much until you find out what's causing the other symptoms, unless she's really hypothyroid and not just a little. Illness can make your body use more thyroid hormone than normal. When she gets better, her levels might return to normal, but if her medication has been messed with too much, she may become really hyperthyroid.
She'll probably have a chest x-ray and/or a chest CT and a lung function test. They may want blood tests, too. The doctor may want to do more, but those are basic tests they'll most likely do.
Why do they want her to see a GI doctor? Does she have GI problems that could cause her lung issues, like chronic acid reflux?
Actually, I just came across an article by Dr. Lowe that describes breathing problems like you describe that ARE attributed to hypothyroidism. I can't post the link now but if you Google "thyroid breathing problems" it should be the first link that you come across. You can also Google "Dr. Lowe thyroid" and see what that gets you.
Jls0711's advice is about hyPOthyroid. I have hyPERthyroid and have associated breathing problems. I am not sure how hyperthyroid causes this issue, but a friend suggested that I may have an enlarged thyroid (goiter) that is restricting my airway. I've had a goiter in the past, but I don't feel anything in my throat this time. However, since your mother does, it could be she has a goiter that is not visible yet. Has she had any tests done on her thyroid to determine whether there is a swelling of the gland?