I would countersink the two existing wall anchors and mud over them, sand and paint to match. Then I would go a couple inches above them install a different type paper hanger (one that the paper slides onto from the end)with a single wall attachment.
Use a metal wall anchor of your choice.

Something like this:
https://www.homedepot.com/p/Gl...20118-2001/204804715

I put one of these by my wife's throne, and it has held up to her many uses a day.
I believe the ease of adding the new roll reduces the wiggle abuse factor a good deal.

Great thread and great advice. Loved the video—can’t believe he actually tested the breaking strength, and 3 samples per item!
I will no longer use the standard toggle bolt that falls to the mud sill if you want to take out the screw.
Very impressed by the strength of the last two toggle styles; with that much strength I don’t care what it looks like if they fail.

The May issue Family Handyman has an article on this topic.