Unfortunately you are not, as you suggest, “the party without the lien”.
Assuming that you and your deadbeat partner purchased this property in cotenancy and the deed of conveyance does not transfer disproportionate interests, then you each own an equal, undivided one half of the whole enchilada.
Consequently the lien for delinquent child is an encumbrance on the debtor’s undivided one half of the entire parcel.
If you two wish to somehow physically carve up the property so that as between the two you have separate ownership, there is nothing to prevent it. But the lien would remain attached to both pieces!
I’m sure that this situation is causing you much grief as it affects the ability to sell or refinance. But as far as the state is concerned there is little danger of it taking steps to enforce its lien. Because the state could only foreclose against the debtor’s undivided one half interest. Your undivided interest would remain in tact and no one is going to make a bid at a lien foreclosure sale to end up owning the other half.
I suppose the state could bid in its lien at such a sale, acquire the debtor’s cotenancy one half interest and then partition a court to have the entire property sold. But that would mean the sheriff selling the property at a public auction and highly impractical.