My problem is that enthalpy is a differential property as I understand it, given up to a constant h0 by
dh=CpdT, i.e., h=h0+∫CpdT |

Not exactly a differential property.

It's just that there is an arbitrary baseline, which is h0 in your equations.

Enthalpy is another name for 'heat content', which is the amount of heat energy you have to put in to assemble the substance from its components plus the heat energy you need to bring to temperature T.

The latter is the specific heat times the temperature change from whatever base temperature you are working.

It is usual to take h0 as the energy of assembly of the molecules from its component atoms minus any energy you get back by forming the bonds.