One problem the OP faces is that in WA the standard for guilt is only the preponderance of evidence on traffic infractions. The destruction of a sign on the side of the road is a indication that something bad happened, quite possibly due to negligence. The OP needs to demonstrate that the accident was due to a flat tire before the accident and depending on the details it could be argued that the flat was caused by the accident rather than the other way around. The OP should check the scene of the accident for tire marks which could corroborate that one tire was flat before the accident as well as document the flat tire. In any case, follow searcher's advice on getting discovery and post it here for review.

If eligible you might consider using a deferral which would not affect your insurance but would likely cost similar to the fine for the infraction.