Ghosthouse, yes, you're right that the Fletcher-Munson Effect will not cause a change in perceived tonal balance if the volume at the listener's ears remains constant.
Regarding your original question, yes it does seem conceivable that changing how the overall system gain is distributed among the various circuit stages that are in the signal path could affect perceived brightness. Although without knowing the details of each of the designs that are involved a more specific explanation would be speculative at best.
For example, Atmasphere has pointed out in a number of past threads that very small amounts of certain harmonic distortion components (e.g. the 5th, 7th, and 9th harmonics) can affect perceived brightness. Changing the signal levels that are processed by the various gain stages that are involved could conceivably affect the degree to which those distortion components are generated. Erik mentioned another example, which is that impedances might change, depending on the specific designs. If the output impedance of a component were to change significantly, and is relatively high, its interaction with cable capacitance could affect treble response to a perceivable degree. And it seems conceivable that changes in impedance at internal circuit points could also affect the treble, depending on the specific designs.
Best regards,
-- Al