This is tough to answer without all the facts.
Class X cores are defined by a minimum knee-point voltage, a maximum excitation current, and a maximum internal resistance.
TPX cores are defined by setting critera for the peak instantaneous error during a specified transient duty cycle and there is no limit in the remanent flux, i.e. the core is not gapped. To do this, many system parameters are required.
TPY cores are also defined by the peak error during a duty cycle, but there remanent flux is limited to a maximum of 10
% of the saturation flux, i.e. the core has gaps.
There is nothing to say that the Class X core will not work with your relay, especially since it seems to be well over dimensioned.
I do not believe that a relay manufacturer can say that the relay must be used with TPY or TPX cores, without laying down any requirements, and for TP cores, the requirements are driven by system specific parameters, such as primary time constants, fault levels, to what degree the fault current is offest, and the duty cycle.
With that said, without knowing your system time constants and without knowing how the Class X core you have will perform in relation to its saturation flux during the duty cycle, it's hard to confirm it will work.
In my experience, many folks specify TP cores without fully understanding the reasons why they are used.
If it were me, I would go back to the relay manufacturer and ask why a TP core is required and what they base that on.
Hope that helps some.