Using Trainable=True in Keras Embedding obtained better performance
It depends upon where do you want to submit your results that you claim, and what is the submission criteria.

First, it is unclear if "lower error in regression" is training or validation/test error. I am sure that letting Embedding layer to be trainable will adapt it to fit the training set better, but it might cause overfitting and cause higher error on validation/test set.

If for your set of tasks were you use Embedding layer you consistently get lower test error, and the requirement does not specifically say so to use trainable=False for Embedding layer - why bother? It is not cheating.