An assembly is provided for a gas turbine engine. This assemble includes a stationary structure configured with a first side, a second side and a bore that extends through the stationary structure between the first side and the second side. The assembly also includes a fluid transfer tube extending within the bore and forming a plenum between a sidewall of the fluid transfer tube and the stationary structure. The fluid transfer tube is configured with a first end at the first side, a second end at the second side, a flowpath and an aperture. The flowpath is at least partially formed by the sidewall and extends through the fluid transfer tube between the first end and the second end. The aperture fluidly couples the flowpath with the plenum.