Section 4.1.3 of the 1980 QFT book by Itzykson & Zuber contains a discussion of stimulated emission for a quantized electromagnetic field interacting with a classical source. However, this is not a full QED treatment since matter is treated classically.
Stimulated emission in full QED can be studied only in the context of bound states or long-living resonance. Bound states and resonances are usually not well-described in textbooks.
Chapter 10 of the 1980 QFT book by Itzykson & Zuber contains a discussion of the Bethe-Salpeter equation for bound states. Weinberg's 1995 QFT treatise treats bound states in Chapter 14 of Volume 1, deriving the Dirac equation (from which one proceeds as you described). On p.560 he remarks ''It must be said that the theory of relativistic effects and radiative corrections in bound states is not yet in entirely satisfactory shape.'' This is an euphemism for ''... is still a poorly understood mess''. See also https://physics.stackexchange.com/a/27623/7924
Today, 23 years later, it is not much better. Probably the main reason is that ambitious people prefer to work on grand projects possibly meriting a future Nobel prize rather than redoing old stuff in a more satisfying way.
More background:
Effective field theories for QED bound states are discussed by Labelle 1998 (preprint at https://arxiv.org/abs/hep-ph/9608491). For QCD, a Dyson-Schwinger approach (to QCD) is studied by Roberts and Williams in a 1994 paper (preprint at https://arxiv.org/abs/hep-ph/9403224), and a semiclassical approach by Hoyer is discussed in a 2011 paper at https://arxiv.org/abs/1106.1420 and (on the tree level) in a 2016 paper at https://arxiv.org/abs/1605.01532.
