If you want real champagne at breakfast, your best bet is to buy a bottle individually to keep in your room on ice. If you are okay with a domestic sparkling wine, the premium wine package Taste of the Vineyards starts at $169 for 3 bottles with options for 5 or 7 bottles at a higher price. Domaine Carneros sparkling wine is listed as included in that package, but they may substitute out due to availability.
You can order a mimosa with your room service breakfast, but it will most likely be made with Celebrity Vintages sparkling wine which is not distinguishable from Korbel by my palate.
Generally you can find Celebrity Vintages or a prosecco at most bars. Some will carry Chandon as well. For the best selection of sparkling wines on the Equinox, go to Cellar Masters wine bar on 4. They should carry Cattier champagne which is covered by a premium package. There are also a couple of upcharge champagnes by the glass at Cellar Masters or if you are dining in Luminae as a Retreat guest.
The Sunset bar (as well as the Magic Carpet bar on Edge Class ships) sometimes carries Veuve Clicquot which you can order by the glass. They made champagne cocktails on our last Equinox cruise in 2019. If you enjoy champagne as much as I do, you might be interested in the specialty Veuve Cliquot dinner with champagne pairings across 5 courses.
Keep in mind that all of this is subject to change, so it's important to be able to roll with the shortages as they affect the onboard selections.