A compressible fluid with wave speed \(c\) lies on both sides of an infinite plane membrane whose equilibrium position is \(z=0\) in a Cartesian coordinate system. The membrane is free to vibrate in response to the fluid pressure, except for two disc regions \(S_0\) and \(S_1\), each of radius \(a\), with respective centres at \((x,y,z)= (0,0,0)\) and \((d,0,0)\). The membrane displacement \(\eta(x,y)\) is constrained to be zero on each of the discs \(S_0\) and \(S_1\), leading to a mixed boundary value problem with different types of conditions according to \(x,y\in S_0\cup S_1\) or \(x,y\not\in S_0\cup S_1\). The system is activated by an obliquely incident plane wave of angular frequency \(\omega\) and acoustic wavenumber \(k= \omega/c\). Asymptotic results are sought in the limits of large \(kd\) and small values of the fluid loading parameter. This is achieved by reducing the problem to a combination of single disc problems, where asymptotic results are known for this simpler class of mixed boundary value problems.