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Lambert has trouble accepting that Aiden is gone, that he will never hear the cat's manic laughter as they cut through a pack of Drowners or see him flip off a dead beast that should have killed him. Lambert has trouble accepting it, so he doesn't.
He works each contract like Aiden is still with him, matching his movements to the cat's. Wearing his medallion alongside his own and wielding his silver sword in Lieu of his original. He doesn't escape into alcohol, doesn't wish himself dead, doesn't do anything that the cat would be mad at him for, choosing instead to honor him with his life. Instead he lives every moment like Aiden would have wanted. Takes on more contracts than even his brothers are comfortable with, and hones himself into a greater fighter than he was ever meant to be.
Lambert has trouble accepting that Aiden is gone, so he doesn't. He finds him among the sounds of battle, the vocalizations that now fall from his lips instead of the cat's. When he closes his eyes and hears metal connect with bone he can feel Aiden's eyes on his.
Lambert has trouble accepting that Aiden is gone, so he wills him into being with every fight. The time between fights simply moments where he goes without seeing him. Until his dying day, when he knows the cat will be waiting for him, Lambert never accepts that Aiden is gone.