We show that asymptotically hyperbolic initial data satisfying smallness conditions in dimensions $n\ge 3$, or fast decay conditions in $n\ge 5$, or a genericity condition in $n\ge 9$, can be deformed, by a deformation which is supported arbitrarily far in the asymptotic region, to ones which are exactly Kottler ("Schwarzschild- adS") in the asymptotic region.