Why would you want to encrypt the CRC byte?

Start to describe - with long flowing text - what problem you want to solve, instead of starting to talk about random details here and there.

A checksum gives a confidence (of varying quality) that a packet isn't damaged.
A signed message gives a confidence (of varying quality) that a packet hasn't been intentionally tampered with.

Computing the CRC of the combination of a secret and your packet data and store in the packet would work as a form of signing. But it would be very weak, since the CRC function is not a one-way trap function, so if I may modify any two consecutive bytes inside your packet data, I can change anything in the packet and still produce the same CRC-16. If I get access to four consecutive data bytes, I can use them to fake any CRC-32 result. And the important thing: I can do it in constant time. No need to test 4 billion combinations.

Now back to you: Do tell what problem you have - not how you think you should solve it.