Don't worry about it. The State doesn't care about rounding error. All you need to do is say "We charge our prices with tax included" - you know, like carnivals and movie theaters.
Then follow the procedures your state specifies for computing reportable tax. Quite likely it wants your pre-tax sales total for the reporting period. To get that, total up your gross sales that you collected, and divide by (1 + tax rate). Just like DJClayworth says, except do it on total sales instead of per-item.
If you need to do the split per-transaction for Quickbooks or something, that's annoying. What Quickbooks says will be pennies off the method I describe above. The state don't care as long as it's just pennies, or in their favor.