Most existing works on fairness assume the model has full access to demographic information. However, there exist scenarios where demographic information is partially available because a record was not maintained throughout data collection or due to privacy reasons. This setting is known as demographic scarce regime. Prior research have shown that training an attribute classifier to replace the missing sensitive attributes (proxy) can still improve fairness. However, the use of proxy-sensitive attributes worsens fairness-accuracy trade-offs compared to true sensitive attributes. To address this limitation, we propose a framework to build attribute classifiers that achieve better fairness-accuracy trade-offs. Our method introduces uncertainty awareness in the attribute classifier and enforces fairness on samples with demographic information inferred with the lowest uncertainty. We show empirically that enforcing fairness constraints on samples with uncertain sensitive attributes is detrimental to fairness and accuracy. Our experiments on two datasets showed that the proposed framework yields models with significantly better fairness-accuracy trade-offs compared to classic attribute classifiers. Surprisingly, our framework outperforms models trained with constraints on the true sensitive attributes.