Is there any way to swap exhaust tips without removing the rear bumper?

If it is the same on the GTC as the GT then yes. You need to remove the rear paneling and try.

I’ve been told the entire rear bumper needs to be removed. I thought there might be a way to unbolt the rear section of the exhaust and pull them out just enough to replace the tips.

I replaced the tips on my GT without removing anything. There is a youtube video too. Get under it or drive the rear on ramps, remove the rear bottom panel and try.

The rear belly is very different from a GT to a GTC.