Lattice enthalpy and ionic charater
Lattice enthalpy is given by the Born-Lande equation:
$${\displaystyle E=-{\frac {N_{A}Mz^{+}z^{-}e^{2}}{4\pi \epsilon _{0}r_{0}}}\left(1-{\frac {1}{n}}\right)}$$
This seems very complicated, but removing some constants, we can reduce it to say:
$$E \propto \frac {z^+z^-}{r_0}$$
here, $z^+$ and $z^-$ are charge on cation and anion and $r_0$ is distance to closest ion.

Coming to your question, In the sulphate series of Alkaline earth metals, the charge remains the same for all members: what changes is the distance to closest ion, which is smaller for small ions such as $\ce{Be^2+}$, thereby causing their lattice energies to be high. 

The decrease in lattice enthalpy may not neccessarily mean a decrease in ionic character. If two lattices have similar values of $r_0$, then a higher lattice enthalpy for one of them may imply larger ionic character due to large charges on either of the atoms.