Goslings often do that so it may be normal unless they’re doing it more often than the other gosling did. Usually with a respiratory illness you might some fluid from the nares which they’ll flick away, sometimes facial puffiness which might be hard to notice, coughing, and sneezing.
Heavy breathing can be an upper respiratory illness but it can also be a lower respiratory infection of the airsacks, or even something that’s unrelated to a respiratory problem, fluid in the abdomen or inflammation of an organ or the digestive tract can push on the lower airsacks and lungs causing open mouthed breathing as can a heart problem.
Dosing medication for a gosling can be tricky because they’re so small, and giving them antibiotics anyway isn’t usually a good idea as they’re at a critical stage where their gut is developing a healthy microbe colony, if you can get ahold of antibiotics I strongly suggest only giving the sick gosling access to any.
Here’s some great advice on Baytril dosing for baby birds
https://www.backyardchickens.com/threads/medications-for-small-chicks-and-ducklings.1441331/
Another option is tylan/Tylosin, it treats a lot of upper respiratory bugs as well as lower digestive bugs.
If it’s possible a vet might be best for this little one considering its age and that it has open mouthed breathing, which is serious even for an adult bird. If that isn’t possible that’s completely understandable and unfortunatly very common, there aren’t a lot of exotic vets, let alone those who are willing or experienced with geese.