I am not particulrly familiar with your application and also it depends on what you consider high (also it depends if you are interested in molar Cp or per kg). Anyway, one idea could be graphite, which has a relatively high Cp and high thermal conductivtiy (preferably oriented in the basal plane direction or even better graphene) at temperatures of 300-400 K. I hope that helps.
If you will integrate the material in such a way that you also need chemical resistance, then you should consider silicon carbide, which has thermal conductivities from 360 W/(m*K) to 490 W/(m*K) at environmental conditions and a heat capacity of 675.4 J/(kg K)
I share you the source of data in the link:
http://www-ferp.ucsd.edu/LIB/PROPS/PANOS/sic.html
I hope this could be useful, it seems you are working in a very interesting project.
Greetings from Mexico.
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