I know little of modern American history so this is just my take...
Indy served in WWI and played a role in WWII I assume but this wouldn't carry much weight with others who had also served. They did their bit as did everyone else.
Obviously Indy stands out more than others but in this paranoid period of reds under the bed, nobody was above suspicion. Accusing fingers were pointed at many upstanding US citizens.
Perhaps also the Fed was a little jealous of Indy's record in comparision to his own?
There have been many times where some in th USA have not supported the role of the armed forces for many reasons, past and present but I would consider the 50's feds would be very pro establishment and therefore unlikely to have an anti war agenda, again especially at this paranoid time.
I don't think the behaviour was wholy inaccurate, merely an overzealous fed, pursuing a line of enquiry, in the belief he is serving his country. He couldn't afford to let hero worship bias his investigation, based on the limited facts he had. It was a reasonable line of enquiry that the FBI would have taken and were too obsessed to pass over just because some old timer had some medals.
It is possible good people turn bad, or loyal people defect.
The guy thought he was doing his job.