Now, if the Sun gets lighter, the planets do drift away so they have more (i.e. less negative) potential energy, but this is compensated by the kinetic energy of particles escaping the Sun... or something.

That's right. The total energy of Sun+planets+escaped matter is classically conserved. Fortunately, the velocities and gravitational fields are small enough for the Newtonian gravity to be a very good approximation, so there are no relativistic complications.

I'm not an expert in general relativity, and I hear that it's non-trivial to define the total energy of a system when gravity is non-negligible, but the local conservation of energy and momentum does still apply.

That's true, the total energy in GR is only defined for a system with an "asymptotic time translation symmetry", but most isolated systems are like that (what happens far away from massive objects is not significantly affected by the details of the orbital motion and such). There is a marginal quality wiki article on the subject.