The recording software seems to install a Silabs CP210x driver so there is a chance that it uses actually a Virtual Serial port. In that case it would be theoretically possible to control that device with the VISA functions in LabVIEW.
HOWEVER: Without a protocol description this is basically useless and the manual does not contain anything that could be even considered an attempt for a protocol description.
So have you contacted the manufacturer? Without a protocol description it is pretty much useless to try to do anything outside of their own control software. And they are the people who can give you such a description, if they want to.