I just toss the whole cuke in the run. Over ripe is good, they love the seeds. It may take them a few days to decide to eat it. With a lot of cucurbits they seem to prefer to eat them when they're somewhat rotted/fermented and mushy.
IMO, if it ain't broke, I'm not going to try to fix it. It's normal for an animal (including human, sounds gross, but true) to have a light parasite load. When it gets out of balance, that's when it's time to cull or treat. Weak animals are more prone to heavy parasite loads. Not to mention, you have the egg withdrawal time, as well as the issue of the parasites perhaps building resistance to the med.
Mine love cucumbers!!!! They eat everything but citrus.