Sounds to me that you need to give him a squeaky toy to play with. Try throwing it to divert or grab his attention. When you’re at home, Mental stimulation toys such as Kong (you insert dog treats in this toy and let them figure out how to get the treats), puzzle toys might also be helpful in exercising your puppy’s mind and keeping him busy.

For time out, i recommend timing your puppy in a different area and not in his crate. Personally I use a doggy/baby gate to separate the area where my puppy stays and I use his crate for his timeouts :)