Hi
As a Bavarian who was born close to Salzburg I need to say that both hotels/ locations are not really a comparison.
Goldener Hirsch is directly in the historic center of Salzburg (Getreidegasse) close to Mozart's birthplace and in the middle of X-Mas markets in December. Fuschl is where I go if I like to have a swim in summer (due to the Fuschl-See) and it's one of the nicest lakes in Austra (and cleanest - but you can drink the water of all Austrian mountain lakes). Having said that, Fuschl is definitely not a hub location for Salzburg (it will take you at least 45min one way to go there, with very limited public transport in between).
As others said, it's either doing both for several days or if you are a first time visitor to Salzburg, I would stay at Goldener Hirsch.
If I go to Salzburg, I stay in the Sheraton - it's closer to the main station, and Getreidegasse can be overrun by tourists so the hotel might be a bit loud. Never stayed there though.
Salzburg is magical during winter and CoVid numbers are extremely low in Austria, so if you can make it to Austria, you'll definitely enjoy.