I don't think there would be much difference because much of the conductivity of copper is from the conduction electrons, not phonons. Isotopic purification increases thermal conductivity in silicon because it decreases phonon scattering.

Isotopically pure diamond, now there's something to look at.

https://en.wikipedia.org/wiki/Isotopically_pure_diamond

"The 12C isotopically pure, (or in practice 15-fold enrichment of isotopic number, 12 over 13 for carbon) diamond gives a 50% higher thermal conductivity than the already high value of 900-2000 W/(m·K) for a normal diamond, which contains the natural isotopic mixture of 98.9% 12C and 1.1% 13C. This is useful for heat sinks for the semiconductor industry."