We prove strong well-posedness results for the stochastic 2D Euler equations in vorticity form and generalized SQG equations, with $L^p$ initial data and driven by a spatially rough, incompressible transport noise of Kraichnan type. Previous works addressed this problem with noise of spatial regularity $\alpha\in (0,1/2)$, in a setting where a rougher noise yields a stronger regularization. We remove this limitation by allowing any $\alpha \in (0,1)$, covering the same range of parameters for which anomalous regularization effects are known to occur in passive scalars. In particular, this covers the physically relevant case $\alpha=2/3$, associated with the Richardson-Kolmogorov scaling of energy cascade.