Is it possible to encode a mpeg2-ts video with a FPGA linux core?
From a performance perspective putting soft processors on FPGAs gets you "the worst of both worlds", you get the inefficiency of general purpose processors (e.g. much less paralellism than custom logic) with the inefficiency of FPGA logic (much lower clock speeds, much higher silicon area per gate than the gates in a "hard CPU").

So soft processors on FPGAs are sometimes useful for control/utility stuff but for heavy number crunching they are a terrible idea. I find it highly unlikely that a soft processor on a FPGA could encode video in realtime. 

A pi isn't going to help you very much either for starters there is no documentation on interfacing custom stuff to the CSI. Also AIUI the only hardware encoder on that chip is for h.264 and the CPU is nowhere near powerful enough to do it in software.