however, this one-way connection means I can't send any information either way. I don't really understand how to connect from both sides to communicate with the program.
As you discovered, you can only have a port open in one application at a time. If you want to use the same port in BenchVue and the Arduino IDE, you need to disconnect from the port in BenchVue before connecting (uploading or using Serial Monitor) in the Arduino IDE. If BenchVue doesn't provide a way to disconnect from the port, you would need to exit BenchVue.
Ultimately, I want to synchronize the multimeter data with the data I am receiving from the Arduino. If there is a better or easier method, I am open to trying something else as well.