Yes, Feeding cochineals can damage the cacti, sometimes killing their host.
You seem to be on the right track, The Cactus Doctor discusses Cochineal eradication. Their recommendations were:
1) A power nozzle attached at the end of your hose.
2) If the infestation begins to get out of control, treating the areas by scrubbing them with insecticidal soap or unscented dish soap was suggested. Neem Oil was also mentioned for a natural approach.
A similar set of solutions were also recommended by the University of Arizona Extension in a publication on Cactus Diseases and in a previous Mr Smarty Plants question
Several webpages mentioned the use of insecticides, and Wikipedia mentioned several natural preditors: “Several natural enemies can reduce the population of the insect on its cacti hosts. Of all the predators, insects seem to be the most important group. Insects and their larvae such as pyralid moths (order Lepidoptera), which destroy the cactus, and predators such as lady bugs (Coleoptera), various Diptera (such as Syrphidae and Chamaemyiidae), lacewings (Neuroptera), and ants (Hymenoptera) have been identified, as well as numerous parasitic wasps.”