I appreciate the suggestions, JKSH.
Answering your questions in order:
What happens if you delete the SV node from your block diagram? Same error code (-1950678943). What happens if you try to write to the variable without opening a connection? Same error code (-1950678943). Right-click the SV node and select "Replace with Programmatic Access". Does this one work? I tried this previously, but it didn't work out. Same error code (-1950678943). What happens if you undeploy everything from your cRIO, and then deploy the variables again? Tried this previously, too. I tried undeploying and then deploying again, but same error code.
What happens if you undeploy everything from your cRIO, and then create a new project with new variables? Nope, same error code. Verified that all shared variables were not deployed prior to creating the new project by using the Distributed Systems Manager.
(Drastic measures) What happens if you reformat your cRIO, install your software from scratch, and try again? Just tried this too, unfortunately no success.
The situation remains as follows
Computer:
Shared Variable Node - Success
Programmatic Access - Success
cRIO:
Shared Variable Node - Success
Programmatic Access - Failure (Error Code -1950678943, Error message: "Timed out while attempting to open a connection to the variable.")