Can any solid material with a low heat capacity exist that feels closer to human body temperature than another solid material with a higher heat capacity; where both materials were previously kept in either a mundane oven or freezer for a sustained period?
Let me rephrase to:
Is there any solid which disobeys the inverse proportionality of thermal conductivity and specific heat capacity?
Consider $1000kg$ of wood and $1000kg$ of aluminium, both at $320K$ (very warm). At the instant you place a finger on such large thermal masses, your perception of temperature comparison is dependent on heat conductivity of the materials, not their heat capacity (their masses are so large compared to your finger, their temperature is almost constant depsite losing heat to your finger). Using such large masses and (equal masses for that matter) is necessary since otherwise I can instantly answer yes to your question by giving you 100g of wood and 1g of gold (beaten to the same surface area of the wood) just taken from the freezer and you would perceive gold being closer to body temperature than the wood after a second. So lets define the question by specific heat capacity, and instantaneous perception of heat transfer.
To answer it though, there is in fact no metal which disobeys this relation due to the electron sea being the majority carrier of kinetic energy in the bulk metal. Their having large mean free paths and low masses allow them to attain very high velocities (which is a property of high temperature) and therefore are able to transfer energy quickly in the bulk material. In other words, if metals used anything heavier to transmit heat, like their nuclei, it would not only take much more heat to accelerate them to the same velocities the electrons could attain (resulting in higher heat capacity), but the rate at which that kinetic energy is transmitted across the material is accordingly slower (lower thermal conductivity). In fact the lattice of metal nuclei do in fact contribute to both properties via phonons not translational kinetic energy like in gases, but phonons are still greatly superseded by the effect from electrons. Therefore the inverse relation between thermal conductivity and heat capacity is valid for metals.
What you are looking for is a non conductor with both higher heat capacity and thermal conductivity than a conductor. For that I give you diamond (figuratively...I can't afford one), which has a specific heat capacity of $0.5 J/gK$, higher than that of any metal denser than vanadium (which is almost all of them), but has a thermal conductivity of $>900W/mK$, trumping silver's $421W/mK$ which is tops for all pure metals.
Indeed, $1kg$ of silver would feel much closer to body temperature than $1kg$ of diamond (that's alot of diamond!) despite diamond having a higher heat capacity.