Effect of specific heat capacity on heat loss
Assume you add a fixed amount of heat to a substance held in an isothermal bath, so that as soon as you start heating it, the substance gives off heat to the surroundings. As heat is added the substance increases in temperature to a maximum. When the heater is turned off, the substance continues to dissipate heat until it is again in equilibrium with the surroundings.
If the rate at which heat is added is much faster than that of heat dissipation and the heat capacity is approximately constant, then the maximum temperature will be approximately
$$T_{\mathrm{max}}=T_{\mathrm{ini}} + \frac{Q}{C_p}$$
where Q is the total amount of heat added. As expected, a substance with a smaller heat capacity will become hotter.
Now consider the rate of heat dissipation. The rate of heat dissipation can be expressed with the heat equation (in one dimension) as
$$\frac{dQ}{dt} = -k \frac{\partial T}{\partial x}$$
where k is the thermal conductivity.
Therefore if k is the same for two substances, then the rate at which heat is released will differ according to the heat capacity, and the lower heat capacity substance will release heat more quickly (since temperature gradients will be greater in that substance).
This can also be argued based on the value of the thermal diffusivity $\alpha$, which is inversely proportional to the heat capacity $c_p$:
$$\alpha = \frac{k}{\rho c_p}$$
As the Wikipedia explains,

In a sense, thermal diffusivity is the measure of thermal inertia.[6] In a substance with high thermal diffusivity, heat moves rapidly through it because the substance conducts heat quickly relative to its volumetric heat capacity or 'thermal bulk'.

Since the substance with smaller $c_p$ will have a larger thermal diffusivity (assuming all else constant) it will also lose heat more quickly.
This can of course be expected to result in greater uncertainty in a measurement of the heat absorbed by the substance.