First, if I have a req for an engineer with a range of $160K-$190K, if you are making $220K I know it's unlikely that you will accept this job.
This is broken thinking, and is an example of the problem here.
I know plenty of folks who've left a $220k job with minimal or crappy benefits, for a $190k job with great benefits. I know others who made $220k by busting their hump at an 100 hour a week place and happy took $190k at a stable large company who was content with 40 hours a week after their baby was born. Salary may be a big motivator, but it's not the only motivator.
Everywhere I've worked the salary range for my "grade" has come out at one time or another. You're not keeping it a secret long term. Tell them the grade range is $160-$190, and ask them if it is worth their time to pursue knowing that. Most people won't waste your time if they really aren't willing to take that pay.