You might be able to do something simple with your Likert-scale outcome values that doesn't require a generalized linear model. A prior square-root or logarithmic transformation of those values might bring the residuals into a form that's good enough for a mixed-model linear regression. Those sometimes help when there is substantial skew toward large values in residuals, as your diagnostic plots suggest.
There can be problems with interpreting coefficients after such transformations. With a binary predictor and a square-root transformation of outcomes, for example, the regression coefficient is in units of "square root of Likert scale values." As Likert scales are already somewhat arbitrary, that might not be a problem for you.
The next simplest approach would be a generalized linear model with a log link. That models the log of the mean values of the outcomes, versus modeling the mean of the logs of the outcomes as you do in linear regression with log-transformed outcome values.
Ordinal regression, another class of generalized linear models, seems more involved at first but might end up being the best choice. You don't have to make any assumption about the form of spacing between consecutive outcome levels provided that they are in order. Ordinal regression can make a lot of sense for outcomes that take a number of discrete ordered values, as yours do. It even can have advantages with continuous outcomes, as it makes no assumptions about the distribution of residuals.
There are two main types of ordinal logistic regressions, proportional-odds (PO) and continuation-ratio (CR) models. With PO you model the cumulative probability of an outcome exceeding some value; with CR you model the conditional probability of an outcome taking a particular value given that the case already has at least that high a level. Choices of link function other than the logit used for logistic regressions are also possible.
Frank Harrell devotes several chapters of Regression Modeling Strategies and his course notes to ordinal regression. The R
ordinal package implements ordinal regression for mixed models.