Hi, coauthor of the Grabby Aliens paper here.
In my view, the correct way to calculate in many anthropic problems is along the lines of the well-explored case of Everett physics: by operationalising the problem in decision theoretic terms.
For the sleeping beauty problem, if one embeds the problem in a repeated series involving bets, and if each bet feeds into a single pot, you arrive at the Thirder position. There is then a consistency argument to make the single-shot problem match that.
Similarily, for the Grabby Aliens problem, consider that civilisations may lodge predictions about the distance to the nearest GC, which can be compared to other civilisations' guesses in the intergalactic council at a later date. Or choose a repeated game in which members of the council reset themselves to the spacetime origin point of a random other GC in the council, by simulation or other method, and make bets from there. The single-shot case, i.e. humanity's predicament, should have a matching strategy.
It is statistical prediction in this sense that I had in mind when helping with calculations+concepts for the paper.