How do I snake this tub drain?
You'll need a different tool to get the plunger and/or linkage - your drain snake would be unlikely to get it. You might have one of these grabber tools somewhere in the house, and it would be worth trying it (down the access plate opening, so it's got a straight shot at the plug.) If that fails, you're probably near the tipping point where calling a plumber might be quicker and cheaper than continued DIY.