Imagine if Hawkeye was the one that got to leave instead of Trapper.
1. Hawkeye also fails to leave a note goodbye, but he leaves his blue Hawaiian shirt. Trapper's yellow one is mysteriously missing. I think rather than going to Kimpo Trapper gets depressed until BJ shows up, and even then it takes awhile for him to warm up. He's a little less willing to open his heart after his best friend leaves. He cheers up when Hawkeye mails him and BJ is a good guy so they do click eventually.
2. Trapper gets so much mail. Hawkeye subscribes him to the Crabapple Courier. He sends treats and jokes. Good booze one day and a dead fish another day. Radar begs Trapper to please reign him in. Hawkeye writes as much as Peg writes B.J. At first B.J. thinks the letters and gifts are from Louise. Trapper replies with something like: my wife has much better taste. My mistress on the other hand...
3. You thought the Trapper Complex was bad. Hawkeye was the chief surgeon and no one EVER shuts up about him both socially and professionally. Even Frank will do it in his own backhanded way. The Hawkeye Complex has its own gravitational field.
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4. Frank amazingly becomes chief surgeon because Potter thinks it makes sense to do it by rank and immediately regrets it. Has to basically fill the role himself until Frank leaves and before Winchester arrives. He gives it to Trapper which immediately causes friction with BJ who thinks Trapper shouldn't get it just because he's been here longest. This is made worse by Trapper being ambivalent and not the least bit competitive.
5. Trapbeej friendship dynamic is way less internal squabbles than Beejhawk, way more torture of Frank. Trapper is kind of surprised that Hawkeye was the voice of reason in retrospect because it did not seem like he was until Frank wakes up on top of the hospital in Klinger's negligee one day. Despite being Frank's main antagonist, Hawkeye also had an equalizing effect on BJ and Trapper. Trapbeej together is like turning up the volume. It's because of BJ's fraternity background paired with Trapper's football background it makes em a little nastier. Rip Frank.
6. Trapper doesn't offer information about himself easily and BJ bottles things up. Trapper is less inclined to push BJ into confrontations, BJ is committed to the bit of being Mr. Nice Guy. They get along well, stick together, do bits, but the few fights they get into are long and nastier than either would have had with Hawkeye. BJ's self-pity over missing his family and taking it out on everyone is even harder to swallow for Trapper who has been there for years and also has a family. Which in turn has BJ mad because of Trapper's chronic cheating seeing it as Trapper not appreciating them and saying things that imply he thinks he loves Peg more than Trapper loves Louise because he's loyal (even with his affair still taking place.) It takes a lot to really piss off Trapper so yeah, I think it gets bad and other parties have to intervene to help them make up.
7. There would be a weird Dartmouth vs Harvard rivalry thing with Trapper and Charles. BJ is surprised because usually Trapper isn't particularly competitive. Charles being from Beacon Hill does not help. The GFA storyline probably has both of them with their names forward as chief of thoracics at Boston Mercy. This isn't to say they hate each other, but it's a side of Trapper that is rarely seen.
8. I just had the thought of Hawkeye starting to write BJ too since he's his replacement. Hawkeye is bored. He wasn't as screwed up by the war as canon but he still watched his childhood friend die on his table. He is trying the small town doctor thing as he works through his issues. His dad starts leaving out travel brochures after two months. BJ is not enthusiastic about his new penpal. He also apparently has no choice because Pierce will write him whether he replies or not. Not even about interesting things, just whatever seems to be on his mind at any given moment, or about the numerous projects he starts and abandons with his free time. He sends the swamp rats one of his insomnia birdhouses. It doesn't have a hole for the bird to get into.
Trapper (fond): you should meet him in real life. Never shuts his mouth.
