While I could care less about the declawing issue . . . It IS A CAT. If you want this to happen again, be ultra-Mr. Sensitive and do a repair again in the future ... Instead, I DO HAVE three solutions which any competent carpet guy can handle.
Discuss this with your apartment manager.
1. They might have carpet remanents for your unit. Use the remanent to repair your floor.
2. Your closet is carpeted. A carpet guy can take a slice from the closet which will perfectly match your existing carpet and install it on the ripped section. He can replace the carpet slice with a part of the remanent they have on-hand. The minor differences will never show.
3. He could remove the closet carpeting entirely and install laminate. Use the closet remanent from the closet to fix the shredded portion.