We investigate the testing-against-independence problem \mw{over a cooperative MAC} with two sensors and a single detector under an average rate constraint on the sensors-detector links. For this setup, we design a variable-length coding scheme that maximizes the achievable type-II error exponent when the type-I error probability is limited to $ε$. Similarly to the single-link result, we show here that the optimal error exponent depends on $ε$ and that variable-length coding allows to \mw{increase the rates over the optimal} fixed-length coding \mw{scheme} by the factor $(1-ε)^{-1}$.