If you just want to get something done, you won't have fun trying to do it with a soft-core on an FPGA, considering you don't have past experience with FPGAs. If you want to try it for educational reasons, that could make sense. As others have pointed out, running a soft-core on an FPGA is a very inefficient use of resources. By just taking an FPGA and throwing a soft-core on it you don't gain much, because you aren't taking advantage of the parallel processing ability of custom logic. A soft-core can be useful as an overseer to manage tasks at a higher level, but many commercial designs don't even do that, they just use an external dedicated processor.
I guess you've decided on using a Due then, which I think is a good choice. It sounds like that should be more than up to the task you will ask of it.