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Bartholemew, who works at the American Embassy, is trying to use lighter fluid to clean a spot off his tie.
"Dry-cleaningwise, things are all fouled-up. I had a good man, a really excellent man on the Rue Ponthieu" Means "I knew a very effective dry cleaner who worked in a shop on 'Pontieu road'"
"H.Q. asked us to use the plant here in the building -- to ease the gold outflow." means "Our bosses in headquarters asked us to use the dry cleaning service provided in-house, to reduce costs"
So "plant" means a dry cleaning service. "Plant" can mean "factory or industrial facility". It's implied that the American embassy is big enough to provide such services. And "ease the gold outflow" is using "gold" metaphorically to mean money.
Bartholomew is choosing, for various reasons, to avoid talking about the important topics and discuss something trivial.