How hard is deciding the existence of a polygonization with prescribed perimeter?
Your problem is NP-hard, since it contains the Hamiltonian cycle problem on grid graphs as special case: Given a set of lattice points in the plane, is there a cycle in which all edges have lengths $1$ (and hence connect two horizontally or vertically adjacent lattice points.
The grid graph has a Hamiltonian cycle, if and only if the corresponding $n$ lattice points (interpreted as geometric points) have a polygonization with perimeter $n$.

Alon Itai, Christos H. Papadimitriou, Jayme Luiz Szwarcfiter:
Hamilton Paths in Grid Graphs.
SIAM Journal on Computing 11, pp 676-686 (1982)