Are we free to use any means necessary, except for sinful ones, to reach the lost?
The problem with saying Yes is that it might omit a category. It introduces the category of sin—"If it's not sin, it's OK"—but there's another category to consider, like wise. And only dealing with the category of sin as the criteria for what you do is probably not wise.
When you ask, "Can I discipline my children in any way that's not sinful?" that's not a helpful way to ask the question. What you want to ask is, "What ways of disciplining or interacting with my children would be maximally helpful to them?" There might be some non-sinful ways that are not very helpful.
So I think I'd answer No to this question.
When you determine that "all things are lawful for me," as Paul says in 1 Corinthians 6:12, and "I have become all things in order to save some" (1 Corinthians 9:22), I think he means "all things" within the context not only of what is not sinful, but also of what is wise, helpful, and strategic.
For example, there are ways of dressing, kinds of music, and kinds of language that you might use which, in and of themselves, aren't in the category of sin. But they might be on a trajectory that would take a church to an unhelpful place that, 10, 20 or 30 years down the road would be hurtful, destructive, or unhelpful.
I have often struggled with the fact that Paul seems to operate with "better" and "best" inside of the category of non-sinful. For example, in 1 Corinthians 7 he says in essence to the single folks, "If you marry, it is not a sin. But I wish you didn't." How can you have an A+ and a B- life, and the B- not be sin? Yet he does. There is a category within non-sinful of preferable and less preferable.
So again, No. I don't think you should operate on the question, "Can we do anything that is not sinful in order to reach the lost."