"there is the obvious aspect of lying to your current employer as to where you are really going"Why is it any of your employer's business where you're going? You presumably have some number of vacation days that you're entitled to as part of your compensation package, so use them. If your company insists on having an accounting of what you do during your free time, they're behaving like unreasonable jerks, so don't feel guilty about lying to them.
"Everyone wins!"
I'm sure that not all companies have dysfunctional cultures that include routinely working on weekends. (Apparently the place where you work now doesn't, since you're free on weekends.) By demanding that they interview you on Saturday, you're asking all the employees who would be interviewing you (which, as you noted, might be the entire company) to make the sacrifice of coming in to the office on Saturday for your convenience. This makes you look like a high-maintenance prima donna, and also gives the company an incentive to ignore you and choose a candidate who is available during the week.