Is the specific heat capacity of liquid iron constant?
Okay, let's answer bit by bit:


Measurement of heat capacity of liquids: this isn't particularly hard to do, and in fact it's much easier to obtain $C_p$ experimentally than $C_V$… however, once you know one, the other can be determined from it and some other thermodynamic properties. Different types of calorimeters are used for liquids, but the best-known is probably the Callender and Barnes apparatus.
“Does an accepted value exist?” and “Is the specific heat of liquid iron temperature-dependent?” — Well, I found a good paper which includes measurements and comparison with previously published values: Thermophysical Properties of Liquid Iron, M. Beutl, G. Pottlacher and H. Jäger, Intl. J. Thermophys., 1994, 15, 1323.

            

which confirms that considering that heat capacity is roughly constant in the 1800–4000 K range. The authors thus extract the constant pressure specific heat capacity as 825 J/kg/K, apparently in reasonable agreement with previous work that they cite:



 
“Or am I a poor researcher?” — Well, the above paper is the first result return by Google for a search for "specific heat" and "liquid iron", so I'll let you be the judge of that ;-)