"Question, so it's not illegal if you have a CHL but can you legally kill a hog with the weapon ... while engaging in archery for deer?"

Use of handgun must be in strict accordance with CHL laws, predominantly covered in Chapter 9 of Penal Code. And there ain't mention in there of shooting pigs.

In my not-so-humble opinion, closest thing applicable could possibly be be "Necessity", aka "Law of Competing Harms". CHL Instructor #6616