(Sorry for my bad english.)
I found a simple soulution for this problem and I am using my Elgato HD60 Pro to record gameplay from my Nintendo Switch.
As I read in your post you already connectet the video to Bandicam.
All you have to do for the audio:
Use an analog audio cable (mine are green on both sites, I added some pictures for better understanding) from your TV (thats what I do) or your console to the blue
audio input on your PC.
Go to Video -> Settings -> Primary Sound Device and select your blue audio input (I don't know the english name, in german it's called "Eingang (...)").
Now your record includes sound!Pictures:
Blue audio input:
Analog audio cable: