How to properly put puppy in time out?

I have a young Keeshond puppy (9 weeks old) and two older cats (ages 3 and 5). He is extremely mouthy, and when I try to play with him, he often lunges at me instead of the toy and bites my hands or arms. His bites are quite hard and have drawn blood a couple of times.

He doesn’t respond to “ow” or other pain signals, so I’ve been using short time-outs instead. After that its crate time, since he only gets like this when hes tired. At the moment, I do this by picking him up, carrying him out of the living room, and placing him in the smaller bathroom. I used to put him in the hallway, but he started chewing on our shoes there.

I’m wondering whether carrying him to time-out is a good idea. I can’t leave the room myself, since the cats are usually downstairs, and the puppy and cats are only allowed to be together downstairs when I’m present.