This is extremely similar to Srednicki Chapter 5 where he includes this picture (but sadly no completion relation similar to your text):  [![enter image description here][1]][1] 

  [1]: https://i.sstatic.net/pDEvl.png

This picture shows the entire set of eigenstates where there is a single vacuum state, a line of one particle states and then a multi-particle continuum. Thus my understanding of your formula is that $|\lambda_p\rangle$ does not necessarily represent a single particle state. In fact, in private correspondence with the author, he suggested I think about the states in this context with one label for momentum and another  label $n$ that captures everything else; this is $\lambda$ here. Therefore, the $\lambda$ sum represents every possible $0$ momentum state and the integral represents the boosted (by ${\bf p}$) counterpart of that state. This way we are capturing the entire basis for the Hilbert space so this is a fine completeness relation.