Wiring in false chimney breast
I don't think it would be a good idea.
It would be vital you to know the possible ambient temperature of the space you are running you're conductors or cabling while the heater was running. Basic apprenticeship instruction teaches that heat has the greatest effect on the function of your circuitry.
NEC Table 310.15(B)(2) shows the multiplier you should use when calculating the current carrying conductors when your ambient temperature exceeds 104F or 40C. Even if you were using 90C insulation if you exceed 80C or 170F you should not use it at all. This would not melt the insulation and is certainly below combustible material levels, but it would effect your circuitry and its ability to carry current.
So if you have no other option but to run new circuits down this chase. You really need to be absolute on the ambient operating air temperature and the types of electrical methods and material you can use to make it safe.
If this sounds like a little much for yourself as an individual, you might need to consult a professional to make the necessary calculations and recommendations.
Good luck and stay safe.