Emissivity depends fundamentally on the temperature, especially for metals. It easily climbs from 0.1 at 300K to 0.5 when hot - for a clean metal.
Since this alloy isn't much protected against oxidation, emissivity depends a lot on the surface cleanliness. If, say, the wire is coloured after a high-temperature trip, it will emit a lot more than if it's blank, for instance 0.5 instead of 0.2.
There would be some nasty details more, like: total emissivity (the most common one) or in some directions and around some wavelengths.
Having checked that, and because emissivity data is scarce, I suggest to pick data from a
similar alloy, like Invar (36% Ni, rest Fe) or
Permalloy. Do not take one containing Cr, Al nor Ti - but FeCo would resemble FeNi.
http://www-eng.lbl.gov/~dw/projects/DW4229_LHC_detector_analysis/calculations/emissivity2.pdfMaybe you hoped to get a two-digit answers... Sorry, emissivities are by nature imprecise, above all for metallic surfaces. Satellite designers do need accurate emissivities, but for that they expose no blank metal: all is anodized, and only perfectly known materials (preferably polymer films or ceramics and glasses) are used as control surfaces, with an accurately controlled cleanliness.