First off, thank you to everyone providing advice and support!
Just to update for others experiencing the same thing as us:
We gave her the 1 cc of calcium gluconate and some Vetrx in the afternoon, ~4pm. Throughout the evening our duck did not appear to get any better or lay anything.
Late that night (~midnight), I heard what sounded like a coughing/squeaking bout that wouldn't stop for like 90 seconds. She was chilling in her pond when it started and then I saw her run and hide into the corner of her house. She was breathing very heavy with a horrible horse noise with every breath. I gave her water in the house and then sat up watching her with our surveillance system til 3am (yes, we have a surveillance system with a camera pointed at the duck house). This duck in particular likes to sit by the window looking out while the other two ducks never do that. I watched her take heavy, full chest expanding breaths all night. In the morning she was soooo lethargic and still noisy, heavy, mouth open breathing. After reading about all the possibilities, I was convinced we had gapeworms.
However, considering that the the recommended liquid Safe-guard dewormer was $30+tax, I decided to call the vet just to see what that would cost.  It was $75 for the visit (for those curious), which I thought was not bad at all. After spending $30 on syringes, needles, calcium gluconate, and vetrx, why spend another $33 on the goat Safe-guard dewormer which could also be wrong after which I would then likely spend $15 on Amoxycillin-clavulanate (that I am lucky to have access to) which also may have been the wrong antibiotic, and then potentially end up with a dead duck along the way?
So, we took her to the avian vet this morning. They said it was likely not gapeworm because that tends to be more severe than what he observed.  Based on hearing noise in her chest, he prescribed a broad spectrum antibiotic (Primor - 120 mg tablets, 1/4 tablet every 12 hours for 14 days) which should help if it's bronchitis, bacterial pneumonia or other likely bacterial respiratory infection. He said he could do a chest x-ray to be more certain (for $170), but I declined that.
In the end, my recommendation to other duck owners who see what I saw in my video would be the following:
We could have had an egg stuck that required either nothing or calcium gluconate, we could have had a bacterial infection, we could have had gape worm, we could have had a viral infection, we could have had a fungal infection, we could have had a cold that relieved itself in 2 days, or we could have had something lodged in her throat. However, I can't tell the difference between them and playing a video may or may not help someone else diagnose. One could easily spend $80 after tax buying the safe-guard liquid dewormer ($30 at TSC), calcium gluconate ($7 at TSC), syringes/needles ($5), antibiotics (~$12 -$25 depending on access), and vetrx ($10) in the hopes of self diagnosing/guessing correctly and feeling one is saving money all along the way. However, if you can't recognize the difference between the potential issues, why not just spend $75 on a vet visit and then $10 -30 on whichever is the correct option? I only say this because, if you guess wrong, and it is one of the more severe issues (like a lodged item or gapeworm), they could be dead in a day before you get a chance to spend $10-30 on your next guess. I suppose the other option would be simultaneously give calcium gluconate, vetrx, antibiotics and dewormer to cover all bases (except fungal, viral or lodged item), but that just ends up saving ~$20 relative to our vet bill/antibiotics (assuming you do not have the items already). For me (having dropped $30 already) the extra $20 for the comfort of having an avian vet physically examining our bird would have definitely been worth it.
Now that I have many of the items (and perhaps a false sense of how to better diagnose which issue we might be having), I might take the give all 4 treatments approach in the future. However, I do that at my own peril...  The only question I would still have is how long to wait before aggressive treatment or a vet visit, but 3 days of worsening symptoms with one awful evening was enough for me this time.
Don't just take my word for it - here is a reference on respiratory issues:
https://www.tandfonline.com/doi/pdf/10.1080/01652176.1998.10807420