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I think the best thing for these two would be to bump into each other in a social setting like the Canteen. Louis is paranoid and in general not all that chatty but if he'd had a drink or two in him, he might be decent at conversation. I mean I wouldn't expect much but y'know ya gotta start somewhere.
Being that, if they 'got along' and Louis saw Eddie a few times he might pick up on the slight homo-erotic tension between the two of them, and use that to his advantage. Louis is kind of expert at drawing out latent homosexuality in others...but it'd get very abusive and intense quickly, so who knows.
Like I say...put them in a thread at the Canteen together, see how their small talk goes like you say, and if they hit it off, expect Louis to be staring at Eddie all creepy around every corner until he decides to act.