I think the dreadful saying "it depends on your specific question" rings true
for this example. Lipid removal will affect 13C more than 15N because lipids
are more depleted in 13C relative to other tissue components and can bias your
sample. Thus I believe that generally people do remove lipids before running
for 13C (or at least make a correction). It is also recommended to use muscle
over whole fish because of reduced variability. These papers might give you
better idea of what to do with your application.

Sotiropolous et al. 2004. Ecology of Freshwater Fish 13:155-160

Pinnegar and Polunin 1999. Functional Ecology 13:225-231

Good luck,