Molecular computing applied in disease diagnosis, personalized medicine, therapeutics, and other applications often relies on picomolar (pM) to nanomolar (nM) range concentration of interacting species, making noise an inherent part of the computation. While the presence of noise in the inputs may work favorably in an artificial neural network (ANN), in its molecular counterpart, namely the Biomolecular Neural Network (BNN), uncontrolled noise may be detrimental as the decision threshold of a molecular perceptron can deviate from the threshold and result in erroneous classifications. To improve the noise controllability in BNN applications, we develop a multi-layer molecular perceptron network that relies on a molecular exchange-based (MEM) perceptron as its fundamental building block. The underlying Chemical Reaction Network (CRN) module used in BNN here also includes negative feedback of the repressor form for additional control over the threshold dynamics. In addition to ReLU behavior, the proposed MEM-based BNN realizes the XOR operation, demonstrating its potential for linear and nonlinear classification in molecular computation.