the M-W10 shows 4 meaning for COPE-- with several different roots.
the fourth Cope is is from the french couper, to cut
to cut (a structural member) to shape a coping or conform to shape of an other member, NOTCH.

that the meaning i associate with a coping saw. it cuts molding to fit against another molding. (exterior joints are mitered, but inner corners are coped.)

the other meaning of cope come from OE, or LL cappa. a head covering --the long ecclesiastical vestment
and meaning 2 to be covered as with a cope (like the top of wall might be coped)
#3 from coupen, MF to strike (which leads to maintain a contest or combat.. and to be able to cope with daily stress)