I think it would change what the meter monitors,
No, it won't change anything.
As I said above, the red lead is a potential lead.
Meter monitors the current in the CT, that's it.
as long as I'm not missing something that might cause an issue
There may be local authority issues if it was wired as in your photo (yuk ) and it was part of a local authority monitored system, but other than that , no change to the way it works.
Just a word of caution with ct's.
NEVER allow the CT to be open circuit as extremely high (lethal) voltages can be generated.
If disconnecting from the monitor device, have power off and short out CT secondary if it is left with the primary in circuit.