I haven't seen the anime, but in the game (which the anime is based off from), Jaeger was defeated by Squad 7 during Chapter 17 where Squad 7 attempts to get back to Randgriz. But on the way, they find's Jaeger unit has barricaded the Great Vasel Bridge while Jaeger now commands the improved Lupus Regnum.
After his defeat, he doesn't return to Maximilian and his whereabouts are unknown. I read that there is a mention of him in Valkyria Chronicles 3 but I haven't played it to confirm. He doesn't fight for the Empire but for the future independence of Fhirald (under Empire control), so having seen the Gallian Militia continuously get the upper hand against the superior might Empire and even not using their new Valkyria as the Empire did, he probably saw that the Empire was not the way for him to gain independence for Fhirald and that it was possible for Fhirald to get its independence on its own.
As for the scene at the end of the anime, it could be the writers forgot about Jaeger's fate in the game, remember it and made a cameo for him at the end. I haven't seen him getting shot in the anime so I'm not sure how much of a possibility there is that Maximilian was mistaken that he was dead.