What are my options in this case, and if I hire an attorney, would I
be reimbursed for this?
You should hire an attorney to negotiate a dismissal of the charges based upon the fact that you did have insurance.
If you had known about the charge and showed up in court on the appointed hearing date, you probably could have negotiated a resolution of this without a lawyer. But, even though your failure to appear was not your fault, at this point it is imprudent to try to do it yourself. There is too much of a risk that you will end up in jail for an extended period of time while the case is resolved if you do that.
You will not be reimbursed for your attorneys' fees. Once you are charged with a crime, your criminal defense expenses are your problem, economically, unless you are indigent, in which case, after you are arrested and arraigned you would have a public defender appointed for you. But, you don't want to be arrested and detained so that you can get a public defender if there is any possible way to avoid that outcome.
Insurance will normally not cover any form of criminal defense representation, even if you are in the clear, although AAA sometimes will post a small bond for you for certain traffic offenses if you are a AAA member.