Copper does suppress a fish's immune system. Bacterial infections are always a threat while treating with copper. The numerous insertion points of the parasites are prime locations for a bacterial infection to set in.
Copper Power is much easier on fish, and the therapeutic range is wider than ionic coppers like Cupramine, and so is much easier to maintain.
The thing about copper is to use small regular doses when ramping up copper levels broken up throughout the day, AM, lunch, PM over 24-48 hours with velvet, or longer 3-4 days with most cases of Ich.
Or even longer if treating prophylactically, especially for copper sensitive fish. Wrasses, Anthias, Angels and a few others. I'll take 5-7 days slowly ramping up for wrasses for example.
Dosing a wide spectrum antibiotic during copper treatment is often a good idea to prevent those associated bacterial infections. Kanaplex is very mild. The "trifecta" of Kanaplex + Furan2 + Metro works for more difficult bacterial infections.
Water quality can also be a problem during copper treatment and ammonia can spike interfering with the fish's ability to transpire O2.
I keep the following in my DT sump soaking in case I need to setup a hospital tank asap: Foam pad for HOB filter and a bag of ceramic media like Fluval BioMax, and a foam pad for a bubble filter, all soaking 24/7 in the sump. Can do the same with bottled bacteria like BioSpira soaking for at least 24 hours.
All those meds/copper can deplete O2 from the water column. A powerhead aimed up at the surface actively rippling the water surface works well.