How to define a complicated function inside the body of Compile?

I want to compile a function in a way to keep its memory footprint down. In the example below, I am trying to compile a function f that makes three calls to bigNastyFunction. I do not want to define bigNastyFunction outside and use option "InlineExternalDefinitions" -> True because then three copies of that function will be inserted into the body of the compiled function verbatim, leading to excessive memory usage. Instead, my strategy is to define bigNastyFunction inside a Module which is inside Compile. My hope is that bigNastyFunction will be appropriately stored as a subroutine, and called by the body as needed.

SetSystemOptions["CompileOptions" -> "CompileReportExternal" -> True];

f = Compile[{{x, _Complex}},
  Module[{bigNastyFunction = Function[{y}, Sin[y]]},

   bigNastyFunction[x] + bigNastyFunction[x^2]^2 + 3 bigNastyFunction[x^3]

 ]

]

The example above doesn't work because Function is not one of the functions that can be compiled by Compile. What workaround is there to define a large (complicated) function inside the body of a Compile'd function?