It took me a while to get my head around the give experience at the end of each session instead of the end of the game - but I came around on it after a while. Duty increases, on the other hand, seem weird in at the end of each session if the team is currently in the middle of a mission. Oh sure, you might have rescued that princess and stolen a crate of blasters, but you've not gotten them back to where they were suppose to go. You haven't gotten your "Hey, good job!" from your superiors yet.

So while mechanically doing Duty Increases at the end of the session makes it fit with the rest of the rewards, narratively it seems a little weird, a bit disconnected from the story.

Am I overthinking this? Should I just just relax and hand it out like everything else?

Personally I'd do end of mission. No reward until they're back out of 'the field' and debriefed.

But I wouldn't give experience in the middle of an adventure/mission, either, unless there was at least several days, preferably a week or more, of downtime that wasn't spent in hyperspace. I think characters need time to reflect on what they've done, analyze what went well and what needed to be done better, in order to gain experience. And spending it on new skills or abilities requires time, too (practicing that new thing enough to have some level of competence).

I do my Duty rewards at the end of the mission, too. How well they planned and executed a mission is a factor in how much Duty they get.

That said, I follow the advice of the Order 66 podcast that I didn't listen to and try to give enough duty every four sessions or so to bump them a Contribution rank. So whether that means more but smaller missions or fewer but bigger ones is up to you.

I give both Duty and XP at the end of each session. That way my players get the (potential) benefits of increased Duty on the D100 check right away, or the next session anyway.