The summary article completely omits the description of how to do the feedback. A laser (and FEL) need a cavity for coherent light emission (typically with mirrors), otherwise we just have spontaneous emission (which is incoherent). So how does one make a gravitational wave cavity, or even a GW mirror?
You might want to skim through <https://en.wikipedia.org/wiki/Astrophysical_maser> which discusses among other things single-pass versus oscillator-based lasers, and whether the lack of oscillating cavity makes an astrophysical source something other than a "true" maser. One would want to apply the same thinking to astrophysical lasers and gravilasers.
In an anti-de Sitter universe (AdS), gravitational radiation eventually reflects off the "screen" and returns to the source, unlike in our universe which as far as we can tell lacks such a screen (and expands whereas AdS contracts). Among other interesting results, this means that in AdS a generic black hole that digests matter in blobs in its early history in AdS never fully evaporates, because the gravitational waves emitted from the blobby growth (as opposed to spherically symmetric infall) reflect back before final evaporation, followed in due course by early Hawking radiation.
In an open universe like ours, probably nothing can outright reflect gravitational radiation (seriously, though, it would be interesting area of research into mechanisms that could generate a "reflected GW" metric). However, if there is a high-amplitude plane-fronted wave with parallel propagation sweeping across the universe, it could conceivably sweep up incident gravitational radiation into the pp-wave and redeposit it onto the sources. Bondi & Pirani did a bunch of work in this area if you're curious about details; one might start with their "sandwich wave". (Nobody knows how to generate a sandwich wave; it would have to simply be a feature of the evolving universe, like the metric expansion of space).
It's a long stretch of the imagination to suggest that this will ever have macroscopically observable effects, and relativity doesn't generally admit any kind of "gravity drive" without exotic matter.
