Means originating from distributions of different variances - DV of interest I want to compare two means (repeated measures) that have very different variances of the underlying distributions (see screenshot below), confirmed by Levine's test.

Since a comparison of the means using a paired t-test or 1-way ANOVA would not be allowed given the violation of the homogeneity of variances assumption, I am wondering whether I can report a statistic that is not about the means but about the difference in variances itself.
For this particular dataset, it is actually meaningful that the cross-subject variability in the second condition is smaller than in the first condition. What would be the most relevant statistic to report?
 A: It is possible to deal with differences in variance in ANOVA using Welch-Satterthwaite approach.
Since you are dealing with what seems like a proportion (whether continuous or discrete) and a ratio (minus 1) there are other choices than ANOVA; for example GLMs or extensions may be able to deal with the variance-related-to-mean that appears to be occuring

Since a comparison of the means using a paired t-test 

Would a paired t-test require equal means? You'd start by taking differences and all you'd need would be that the differences had the required properties.
Are the data actually paired?

whether I can report a statistic that is not about the means but about the difference in variances itself.

Certainly. For that matter, even if you couldn't test the means (which I believe you can, if you have a suitable procedure), you could still report a statistic (such as the difference in means).

For this particular dataset, it is actually meaningful that the cross-subject variability in the second condition is smaller than in the first condition. What would be the most relevant statistic to report?

We can't tell you what is most relevant for your purposes. A ratio of variances might be reasonably suitable as a statistic to report, but if you wanted to use a hypothesis test, I wouldn't use the corresponding ratio-of-variances test. Maybe Levene's test or Browne-Forsythe or something along those lines (but if there really is some kind of pairing present I'd want to understand that before suggesting anything).
