I also found answers to my questions in the work of Gehrke & colleagues. Gehrke & Priestley published a paper titled "Canonical extensions and completions of posets and lattices" which answered parts of my questions, in particular, the one I so poorly described as the beta-compactification-like question. Proposition 2.1 and their discussion presents compact completions with the universal mapping property, and they show where the MacNeille completion fits in. The G&P paper is strongly related to a 2nd paper by Gehrke, Jansana, and Palmigiano in which these ideas are also illustrated in detail.

Matthias, I was reading research by Twuenissen and Venema (staff.science.uva.nl/~yde/papers/mcnle.pdf). I had not found research describing "sizes" of completions,so I look forward to reading Erné and understanding what you mean by invectives objects and that the D-M is an injective hull of a poset. TY.