First law is conservation of energy. It states whatever energy goes in or comes out of the system must have the same change in internal energy. But shouldn't conservation of energy apply on total energy (i. e,Enthalpy the sum of internal energy and pressure - volume energy.)?
The change in "pressure-volume energy" you're talking about is just the $p\, dV$ work, which is already included in the First Law $dU = dW + dQ$ in the $dW$ term. So the First Law holds true.
If you want, you can work with the enthalpy $H$ instead of the internal energy; all this is doing is effectively moving the $p\, dV$ from the right-hand side of the First Law to the left-hand side, i.e. pulling it from $dW$ into $dH$. Similarly, you can define many alternative thermodynamic potentials that absorb any kinds of work terms you want.
However, these are all less fundamental than the internal energy, and their utility depends on the context. So we prefer to phrase the First Law in terms of just internal energy.