Advice:
(defun lsp:completion-options-trigger-characters?@remove-quotes (orig-fn obj)
  (let ((chars (funcall orig-fn obj)))
    (if (nth 3 (syntax-ppss)) ;; point at string
        (remove "\"" chars)
      chars)))
(advice-add 'lsp:completion-options-trigger-characters?
            :around #'lsp:completion-options-trigger-characters?@remove-quotes)