No, we cannot have $\int_{-T}^T |\widehat{\mu}|^2 \lesssim T^{1-s-\epsilon}$. This would imply that $$ I_{s+\epsilon/2}(\mu) = \int d\mu(x)\int d\mu(y) |x-y|^{-s-\epsilon/2} = c \int |t|^{s+\epsilon/2-1}|\widehat{\mu}(t)|^2 \, dt < \infty ; $$ to see that this last integral is finite, cut it into dyadic pieces $|t|\simeq 2^n$. This in turn gives the corresponding Cantor set $C$ positive $s+\epsilon/2$ Riesz capacity. This is impossible for a set of Hausdorff dimension $< s+\epsilon/2$.

(The Riesz capacities behave just like the Hausdorff measures, they switch from being positive to zero at a number that one can call the capacitary dimension, and this is always $\le \dim_H(C)$. A good source for all this is Mattila, Geometry of sets and measures in Euclidean space.)

No, we cannot have $\int_{-T}^T |\widehat{\mu}|^2 \lesssim T^{1-s-\epsilon}$. This would imply that $$ I_{s+\epsilon/2}(\mu) = \int d\mu(x)\int d\mu(y) |x-y|^{-s-\epsilon/2} = c \int |t|^{s+\epsilon/2-1}|\widehat{\mu}(t)|^2 \, dt < \infty ; $$ to see that this last integral is finite, cut it into dyadic pieces $|t|\simeq 2^n$. This in turn gives the corresponding Cantor set $C$ positive $s+\epsilon/2$ Riesz capacity. This is impossible for a set of Hausdorff dimension $< s+\epsilon/2$.

(The Riesz capacities behave just like the Hausdorff measures, they switch from being positive to zero at a number that one can call the capacitary dimension, and this equals the Hausdorff dimension. A good source for all this is Mattila, Geometry of sets and measures in Euclidean space; see Theorem 8.9 there for this last statement.)