Is there any harm in simply including all four candidate predictors, and seeing which are significant using the "lmerTest" package?
Breaking that into two questions, the answers are respectively "no" and "yes."
Separating out the different predictors throws away shared information and can lead to problems like omitted-variable bias. Sometimes the effect on one predictor might depend directly on the value of another, or the influence of one predictor might best be seen when other predictors are also taken into account. The only harm in "including all predictors" is when you have a low ratio of cases to predictors, which is not your problem here. Even then, there are principled ways to proceed that effectively penalize predictors to avoid the overfitting that would otherwise ensue. So there's no harm in including all 4 of your predictors at once.
You should, however, reconsider the part about "seeing which are significant using the 'lmerTest' package." If your interest really is in prediction and you can expect to have values for all 4 of your predictors in future cases, there's no reason to cut back on the number of predictors. That will just tend to make predictions less reliable. In general, for prediction you want to include as many predictors in your model as is consistent with not overfitting. Finding a predictor "not significant" might just mean that your study was too small to document its effect, and if your model is used at a larger scale it might be seen as quite important. Statistical significance is not to be confused with practical importance, in either direction.