Low within-scale variances leads to odd distribution

#1
Hello everyone:

I'm a first time poster and I believe that this thread belongs in the General Stats topic, but feel free to correct me if you think it would have a better home under another topic.

I have a nine-item measure with four response options. Across ~450 participants, ~180 had zero (0) within-scale variability for this measure. Although some of these participants were removed after data cleaning (e.g. multivariate outliers, no variance across all scales), ~130 participants with zero within-scale variance remain.

What this leads to is what I might call a quad-modal distribution. Essentially, around 20 people endorsed all 1s and 4s (so 40 total) and around 50 endorsed all 2s and 3s , with scatted values in between these peaks (essentially, it looks like four fingers extended).

I can only think of three options, none of which are ideal:
1) Remove the individuals who have zero within-scale variance: This doesn't seem appropriate to me because zero within-scale variance for one measure alone doesn't necessarily indicate careless responding. Also, this would kill my power to detect an effect (This variable is a mediator in a moderated mediation model so large sample size = good).
2) Model them as separate distributions with a split at 2.5: Although this looks slightly more normal, there are still issues at each tail, the high mode is odd, and I can't think of a way of doing this without categorizing the splits (e.g. 0=low, 1 = high), which would lose some explanatory power as a categorical variable.
3) Transformation: I can't really think of a way to transform these values to make them resemble a normal distribution.

My questions:
1. Can anyone point me in the direction of relevant literature (I have searched, but to no avail) regarding handling zero within-scale variance?
2. Which option above (or other option) is most ideal for dealing with this distribution?

Thanks in advance!