What is a quasi-Hopf algebra filtration? I know what a Hopf-algebra filtration is, but what is the extra condition needed on the Drinfeld associator $\Phi$? And given such a filtration how does one define the corresponding associator on $\text{gr}H$? I cannot seem to find any reference explaining this, so presumably it should be easy.
However I know some examples of quasi-Hopf algebras that are radically graded, i.e. $H=\text{gr}H$ w.r.t. the Jacobson radical where $\text{gr}\Phi=\Phi$, that is $\Phi$ is itself graded in some sense.
I am not sure if the following remark answers your question, but if $H$ is a finite dimensional quasi-Hopf algebra and $I$ is the radical of $H$, then if $\Delta(I)\subseteq H\otimes I+I\otimes H$ i.e. if $I$ is a quasi-Hopf ideal, then the filtration of $H$ by powers of $I$, is a quasi-Hopf algebra filtration. Thus, the associated graded algebra $gr(H)$ of $H$ under this filtration has a natural structure of a quasi-Hopf algebra. In this paper, the class of radically graded, complex, finite dimensional quasi-hopf algebras whose radical has prime codimension, is investigated. There are interesting references therein, among which I believe that this one may be particularly helpful to your question.
Hope that helps a bit.