Speedy trial rules apply in criminal cases, not traffic infractions. You are supposed to have a hearing within 120 days of the citation unless the court, for good cause, has continued the trial. You may make the motion but you are likely to have a problem.
Normally, if you show that you purchased the tabs before they expired and that they are now on the vehicle, the court (at least in Pierce County) dismisses the ticket with costs. It is not a moving violation so would not affect your insurance.