(If yes, is there a reference for this statement?)

$\begingroup$ Addendum: I also wanted to know whether there are known examples of unitary modular tensor categories that do not have a Lagrangian algebra but do have zero central charge $\endgroup$ – Frank Feb 9 '17 at 21:03
The Drinfeld center of a spherical fusion category has topological central charge $0\pmod 8$ see Remark 5.19 in
Müger, Michael: From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180 (2003), no. 12, 159–219.
There are modular tensor categories with central charge 0, which are not the Drinfeld center of a fusion category.
Take the pointed modular tensor category $\mathcal C(\mathbb Z/{17}\mathbb Z,q)$ with $q(x)=\exp(16\pi i x^2 /17)$. It is easy to check that the metric group $(\mathbb Z/{17}\mathbb Z,q)$ has no Lagrangian subgroup, thus no Lagrangian algebra. The easiest way to see it is that $17$ is no square. This category is $\mathrm{SU}(17)$ at level 1 thus has central charge 16 which means topological central charge $0\pmod 8$. Another way to calculate the central charge is by calculating the Gauss sum.