How should I prepare a bathroom subfloor before installing snap-together vinyl tiles?

Yes, the new plywood will cover a multitude of troubles. Your plan is a good one, and I can offer a couple of suggestions for getting the best results.

First, try using a hand-held electric power planer for removing that high seam, instead of a sander. A belt sander will do the job, but it’s much slower and more dusty than a planer. Either way, you’ll need to drive nails or screws below the surface before you start. Getting the old subfloor flat within a couple of millimetres or 1/16-inch will be fine. The old glue that’s on the floor won’t matter as long as it’s only a millimetre or so high. A shovel with a sharpened tip is effective for scraping off anything higher.

When you put the new plywood down, use screws and construction adhesive to secure it. This is essential for squeak-free results. Aim for screws every 6 to 8 inches along all joists. The 3/8″-thick plywood you’ll be adding will easily span small gaps and cracks in the old subfloor, so don’t worry.

Just one more thing: rigid, click-together flooring is quite dependent on a very flat subfloor. The vinyl you’re putting down is probably flexible, but if it’s not, check the manufacturer specs for the maximum dish or crown that’s allowable in the prepared subfloor.