Indeed he would - because Paul knew full well it is still "a sin" - to take God's name in vain -even for Christians.
Rom 7:
7 What shall we say then? Is the Law sin? Far from it! On the contrary, I would not have come to know sin except through the Law; for I would not have known about coveting if the Law had not said, “You shall not covet.” ...12 So then, the Law is holy, and the commandment is holy and righteous and good.
13 Therefore did that which is good become a cause of death for me? Far from it! Rather it was sin, in order that it might be shown to be sin by bringing about my death through that which is good, so that through the commandment sin would become utterly sinful.
14 For we know that the Law is spiritual, but I am fleshly, sold into bondage to sin. ...16 However, if I do the very thing I do not want to do, I agree with the Law, that the Law is good. 17 But now, no longer am I the one doing it, but sin that dwells in me. 18 For I know that good does not dwell in me, that is, in my flesh; for the willing is present in me, but the doing of the good is not. 19 For the good that I want, I do not do, but I practice the very evil that I do not want. 20 But if I do the very thing I do not want, I am no longer the one doing it, but sin that dwells in me.
21 I find then the principle that evil is present in me, the one who wants to do good. 22 For I joyfully agree with the law of God in the inner person,
Paul says that a lost person does not submit to the Law of God - neither indeed CAN they.