There are newforms of weight 2 and level 1024 (and trivial nebentypus), e.g., 1024.2.a.a. The associated abelian surface has conductor the square of the level, so $2^{20}$ (and endomorphism ring $\mathbb Z[\sqrt{2}]$). So the bound 20 for the conductor exponent is sharp, even for abelian surfaces with RM.

(Edit: answer below was given before the QM assumption was added)

There are newforms of weight 2 and level 1024 (and trivial nebentypus), e.g., 1024.2.a.a. The associated abelian surface has conductor the square of the level, so $2^{20}$ (and endomorphism ring $\mathbb Z[\sqrt{2}]$). So the bound 20 for the conductor exponent is sharp, even for abelian surfaces with RM.