I think you're going to have to provide code in that scriptblock to run a self-elevating script. Even then your users will receive a prompt from UAC.
Rather than use Start-Job (which is convenient) you may have to use Start-Process with the "-Verb RunAs" and "-Passthru", store the result in a variable, and then check the state of the process periodically to see if it's still running. The problem with that is that you cannot use the "-RedirectStandardOutput". That probably (I haven't tried it) can be overcome by redirecting the command's output to a file in the typical DOS fashion.
Another problem that you'll probably encounter is that you won't be able to run chkdsk on the c: drive because it's locked another process. The user may be prompted to agree to scan the disk on the next system restart, but that defeats your intent to not inconvenience the user by waiting for it to complete.