I realized that enthalpy is defined as the total "energy content" of the system. Given that in Hamiltonian mechanics we also deal with the total energy H = T + V, can we somehow use Hamiltonian mechanics (or some modification of it) to solve thermodynamic problems?
For eg, H = U + PV, can this equation be dealt with Hamiltonian mechanics?
Lagrangian and Hamiltonian mechanics requires holonomic constraints, and non-conservative effects like friction introduce non-holonomic terms. There have been some clever ways to rephrase friction forces without these terms in special cases. But general purpose thermodynamics has been tricky.
There was a clever approach described in The Classical Mechanics of Non-Conservative Systems (Galley, 2013) which used a Green function and two different paths through time (forward and backwards) to solve problems with general non-conservative effects. This approach was sufficient to permit construction of Hamiltonian solutions for problems such as viscous flow. It is likely capable of answering the thermodynamic problems you seek.
Mechanics (it doesn't matter if Newtonian or Hamiltonian) and Thermodynamics are different theories, working at different levels of description of the physical world. Different doesn't mean they are entirely unrelated, but the relation is not an equivalence. A clear signal is that classical Thermodynamics was already an established theory a few decades before the atomic hypothesis was universally accepted in Physics.
The conceptual bridge between the mechanical description of systems made of many microscopic constituents and classical Thermodynamics is Statistical Mechanics. Through such a theory, one can obtain all the constitutive equations, fundamental equations, or equations of state required to apply the basic principles of thermodynamics to real systems. However, notice that besides a few prototypical problems related to non-interacting systems, nobody directly solves typical equilibrium thermodynamic problems using statistical mechanics. It would be a waste of resources. Once we know the conditions ensuring a typical thermodynamic behavior of a Hamiltonian system, we use Statistical Mechanics to evaluate the specific form of enthalpy, free energies, or equivalent fundamental equations for the system of interest and use the thermodynamic formalism to solve any thermodynamic problem for that system.
As an example directly related to the final question, Statistical Mechanics allows us to evaluate (almost always numerically) the enthalpy of a Hamiltonian system of many interacting particles at a specific thermodynamic state. It can be done using one of the many numerical simulation methods (within the two big families of Monte Carlo and Molecular Dynamics algorithms). Once we get the values of enthalpy for different states, we can use that function to study and predict that system's thermodynamic behavior (including phase transitions).
