[...]causing the structure to start to drift[..]
s/causing/allowing
(The way I've always thought about it is that you'd be setting it up as a nonstable (i.e. not stable, not
unstable) equilibrium. Not strictly an energy minima, maxima
or inflection, but an isolated plateau, as by the time you get to the edges of this flat part of the graph, you've got the shell touching the star (and vice-versa) and other problems to deal with. Should be 'easy' enough to keep nudging it to get rid of perturbations, if you're already a shell/swarm-building civilisation. Or would the pressure of the incident solar winds already be able to help you out?)
How far away is the thing anyway? Could we just hurl a probe at it and find out in a few hundred years?
It would take several orders of magnitude longer than that...
To put some hard figures into it, the star is 1,480 light-years away. That's 1480 years it would take for the probe's
signal to get back to us. But first you need to get the probe there in the first place.
Something launched similar to Voyager 1 would (at its current speed) take over 26 million years to get there, if I've not slipped an order or two up or down, by accident. (New Horizons would be ~34 million years, but then I suppose they wanted to dawdle as much as possible whilst passing Pluto.)
We could probably halve those times, if not more, but it's still millions of years (plus 1,480). Meanwhile, better telescopes and/or extraordinary scientific breakthroughs in space-transit could make it a redundant mission. (Or the demise of the human race, by its own hand or otherwise.)