Is enthalpy of a reaction meaningful at variable pressure?
Why should the enthalpy of a reaction be defined at constant pressure? Why can't it be defined at variable pressure?


Not sure "should" is the right choice of words: It can be defined for a reaction either at constant volume (variable pressure) or constant pressure (variable volume). You might choose one measurement over the other, but you'll get two different quantities.

At constant volume, there's no work done by the system on the surroundings, so you're just getting the change in internal energy of the system (this is typical of a bomb calorimeter lab setup); in the other case, you get the internal energy change as well as the work, and that is how enthalpy is defined (mathematically). See this Wikipedia article for a little more detailed discussion.