Intermittent no start, I changed the starter battery ignition switch?

It sounds like the BCM or NATS antenna amplifier is going out which controls the vehicle immobilizer. To be sure we need to do a CAN scan. You can get a CAN scanner (Controller Area Network) from Amazon for about $36.00.

Here is a video to show you how:

https://youtu.be/u-4syLc-ifQ

Here is a scanner link:

https://amzn.to/3eCqWWw

Here is the immobilizer wiring diagram so you can see how to works and how to replace the BCM in case you need it after the scan. The BCM will need to be programmed which the rebuilder will need to know your VIN number. Check out the images (below). Please let us know what happens.

Can the keyless entry antenna in the trunk cause this? I noticed it was hanging loose in the trunk and I sometimes hear a snapping sound like an electrical shock like there's a short from the back of the car.

Yes, that can do it, can you upload an image of what you are seeing? Here is information on the keyless entry receiver. Check out the images (below). Please let us know what you find. We are interested to see what it is.