You can integrate happiness over the proper time along those world lines; I suspect that's equivalent to integrating a happiness density that looks like SUM_i h_i(t) delta(x - x_i(t)) over spacetime.

Ah. It's about time the assumptions were made clear. I thought that 'creation of happiness' was a function defined on spacetime, and the proposed definition defines the total happiness to be only the happiness created on the observers world line. I believe this is not Lorentz-invariant - while a scalar H(x,t) might be invariant under such a transformation we are interested in H(x,t)dt, which messes up the invariance. And I think your remark about the determinant is just a rewording of my point: the determinant of a matrix describes the change in volume of a (in this case) 4-dimensional volume, but if we integrate only in one direction our result can still change (almost) arbitrarily. And therefore introducing an all-space integral solves the problem - this quantity does deal with all four dimensions.