There's even in-canon precedent for this kind of Indy adventure. In Kingdom of the Crystal Skull, we learn that once the U.S. entered World War II, Indy became an Army intelligence agent and was eventually decorated with a ton of medals. He served in the Pacific Theater. He was a double agent in Berlin. Indiana Jones was a secret agent who infiltrated the Nazis! He was the American James Bond, which is what George Lucas wanted him to be to begin with. Why can't we see some of that happen?
For three movies Indiana Jones is an abnormally badass archaeologist, and then we cut to the fourth movie and find out he was once an appropriately badass soldier. Let's see Indy do all his classic Indy things with American, British, or Russian troops at his side. Let's watch him become the badass spy/soldier/archaeologist/war hero we're told he becomes as he dodges mortar shells and helps to defend or attack a city, all while he races to find the Holy Flip-Flops of Christ before they fall into Nazi hands.
By now, Harrison Ford is a little too old to play a slightly younger Indy than when we last saw him. That's not a problem. If we can erase Kingdom of the Crystal Skull from our minds, we can easily erase a tiny age discrepancy.