Solution
Unfortunately, it is not. The LabVIEW FPGA Module does not have the capability to retrieve a signature or checksum of a loaded bitfile.
The workaround NI recommends is to write a constant to the front panel of your FPGA VIs using an indicator. Then, from the host VI, you can read the constant on your bitfiles and compare it to the expected constant of a given bitfile. If they match, you know you have the correct bitfile loaded. If they don't, you can make the appropriate decisions given your application.
To read indicators of an FPGA VI, see the following article:
Reading FPGA VI Indicators (FPGA Interface).