Can Bentonite Clay expedite the tattoo removal process?

A fanciful idea, at best. And quite an excellent question.
Yes, tattoo pigment contains a positively-charged metal atom, but it's part of a stable mineral salt crystal with no net charge. The crystal is insoluble (thank goodness, because most of the heavy metal ions would be toxic, otherwise.) The crystal's inert nature is the very thing that makes it so suitable for tattooing. I think this is all the chemistry we need to know to evaluate claims about clay tattoo removal, at least as you presented them in your question.
But let's probe a little deeper, anyway. Friends of calcium bentonite claim that its "natural" negative charge attracts the "positively-charged" ink (a uncharged chunk of mineral crystal, remember) out of the skin. Taking the example of a mercury-based ink (yes, they exist), a mercury ion has an electrical potential (think of it as the "attractiveness" it has to the clay) that's super, super, super tiny. There's a lot of tough skin between it and the clay. If it WERE charged (remember, it's not) the attraction would be so weak that they might was well be miles apart instead of 1 or 2 millimeters. To put it in terms of magnetic force, think of using a refrigerator magnet to pull a paperclip through a brick wall, and that's how effective the plan is.
It's important to add here... Laser energy doesn't break the ionic bonds of the crystal, it just breaks the crystal into smaller pieces for easier disposal. Clay still wouldn't work when combined with laser treatments.
So...
Laser tattoo removal continues to be the most effective option. I'm very open-minded about alternatives and would love it if there were a cheaper option that's at least as safe and effective as laser treatment.
Thank you for posting your question!