A derivation is here:
or in Landau-Lifshitz. Bethe's original derivation is found e.g. in Matt Schwartz's Harvard lecture here
The leading contribution to the Lamb shift is the one-loop level (the first non-classical correction) but of course, the effect receives corrections at every higher order, too.
One may derive it in the operator approach or path integral approach, much like pretty much everything in physics. These are just equivalent languages to do physics.
The Lamb shift has to deal with an atom which is not quite an elementary particle. So the usual perturbative rules of QED have to be "generalized" to deal with the composite object. However, otherwise it's about a virtual photon emitted and reabsorbed by the atom. If the atom were elementary, it would be a simple "photon loop" correction to the atom's propagator. A divergent term has to be removed – equivalently, one has to find out sensible limits of the integral – and what is left is some "truncated logarithmic divergence" that produces those 1,000 MHz for the relevant levels.