Short answer: Yes, the carcass is still kosher.
In the laws of shechita it is mentioned that it's possible to make an animal treif even during the shechita process.
See Yoreh De'ah סימן כו - דין נקב בושט או בבני מעים קדם (גמר) השחיטה
א: נִקַּב הַוֵּשֶׁט בִּשְׁעַת שְׁחִיטָה כְּנֶגֶד הַמָּקוֹם שֶׁשָּׁחַט כְּבַר, טְרֵפָה
ב: שָׁחַט הַקָּנֶה וְנִקְּבָה הָרֵאָה קֹדֶם שֶׁשָּׁחַט הַוֵּשֶׁט, אוֹ שֶׁשָּׁחַט הַוֵּשֶׁט וְנִקְּבוּ בְּנֵי מֵעַיִם קֹדֶם שֶׁשָּׁחַט הַקָּנֶה, טְרֵפָה
But you asked
But what is if the injury happened after the shekita?
Once the shechita is over, there's no way to make a kosher animal treife - i.e. make the shechita non-kosher.
However, in the next Siman סימן כז - שלא לחתך אבר מהבהמה בעודה מפרכסת you will see that one may not yet cut pieces off a slaughtered animal that is still moving - and if one does, one may not eat them until the animal is motionless - and really dead.
א: חָתַךְ מִבְּהֵמָה לְאַחַר שֶׁנִּשְׁחֲטָה כָּרָאוּי וְעוֹדֵּנָהּ מְפַרְכֶּסֶת, אָסוּר לֶאֱכֹל מִמֶּנָּהּ עַד שֶׁתָּמוּת הַבְּהֵמָה
However, cutting off pieces will not make the animal treif, since one can eat them after the animal dies.
So we see that the answer is - as you suspected - that the animal is not treif.
As to your specific example of a cat: The Talmud in Horios 13b mentions that eating where cats and mice have nibbled could be bad for your long-term memory.
This is brought down in Halacha - for example in Kitzur Shulchan Aruch 32
לֹא יֹאכַל מִמָּקוֹם שֶׁאָכַל עַכְבָּר אוֹ חָתוּל, דְּגַם כֵּן קָשֶׁה לְשִׁכְחָה
So one should probably cut away around the missing piece.
And then there's the health hazards; one may not do anything dangerous, and eating contaminated food would qualify. But I suspect that's not within the scope of your question.
BTW: Your statement
a carcass is not kosher if it was injured by anything before the death is not 100% accurate. Only certain injuries make an animal non-Kosher.
Which? Those that our sages determined to be life-threatening, usually defined as the animal probably won't live another year with such an injury.