Assuming your arborist's evaluation of the destroyed trees is accurate, you do appear to have a valid claim for just under $10,000. To pursue the bond and liability insurance, you would typically need to file a lawsuit in Superior Court. However, sometimes we are able to work directly with the contractor's liability insurer to procure a payment for our clients in such circumstances.
Adding to David's answer, Washington has a statute that provides for treble damages (in your case $30,000) for "timber trespass," but that statute only applies when the trees are wrongfully cut down. It seems highly doubtful that your circumstances would qualify. You may have heard about treble damages for cutting down your trees, but the statute was not intended to apply to your kind of situation.
You can verify the status of the contractor's bond at the Washington Labor and Industries site (link below). That also will tell you the amount of the bond and whether there are any claim against it.
The amount that you requested seems reasonable, but it no an exact fit for the damages that you might recover in a lawsuit. To get a clear answer, you would need to go over the actual facts with an attorney. You may be entitled to more compensation than you are requesting, even if collecting it would be difficult.
As a starting point, check your paperwork. You should have a bid or a written contract. It usually will have an attorney fee provision and may have an arbitration provision. I would start with the paperwork you then take it from there, If the bond has no claims against it, your chances to be compensated look good.
This is legal information and not legal advice. The final answer to your question will depend on more facts than you can include in your question and some that you probably would not think to include. Treat this as a starting point, not the answer.
Some people would be glad they got three mature trees removed for free. My understanding is that the cost to remove a mature tree can be expensive. Perhaps this is one way to look at the situation if you are not too attached to the trees.
An issue may be whether these trees should be removed soon any way. That may reduce your damages.
The tree service may be licensed by the State of Washington. But, it definitely is not "insured by the state of WA".
You can review the information provided by Labor & Industries to have an overview of what you need to do to make a claim against the bond: http://www.lni.wa.gov/TradesLicensing/Contractors/HireCon/Problems/default.asp .
You can also hire an attorney and not have to worry about learning what to do.