The answer to your question is no, nobody has used persistence to improve the algorithmic efficiency of computing differentials, although of course the relationship between persistence intervals of a filtration and various terms in its Leray spectral sequence have been [described rather explicitly][1] by Basu and Parida. 

Here's an elementary observation: as mentioned in that article by Carlsson and Zomorodian, every sequence of $k$-modules admits a straightforward reinterpretation as a graded $k[t]$-module where $t$ acts by moving things forward one step along the grading. The existence of a persistence barcode relies crucially on the structure theorem for graded modules over graded PIDs. Since $k[t]$ is a PID only when $k$ is a field, relying on persistence will not solve any extension problems for you when you try to compute differentials -- all your $E_{\bullet,\bullet}^\bullet$s will have to be vector spaces already. 


  [1]: http://arxiv.org/abs/1308.0801