  We initiate an investigation of lattices in a new class of locally compact
groups, so called locally pro-$p$-complete Kac-Moody groups. We discover that
in rank 2 their cocompact lattices are particularly well-behaved: under mild
assumptions, a cocompact lattice in this completion contains no elements of
order $p$. This statement is still an open question for the
Caprace-R\'emy-Ronan completion. Using this, modulo results of Capdeboscq and
Thomas, we classify edge-transitive cocompact lattices and describe a cocompact
lattice of minimal covolume.
