Having been a couple of years in a factory that cutted sheet metal by laser, I must say that scanning over an silk screen would be nonprofitable work to a machine which is designed to split steel. You can cover maybe one square centimeter at the same cost that is needed for one meter long slit.
A specially constructed low power machine would be a better choice - something that can splice only thin plastics and that machine just now hasn't something to cut. To increase the speed some unfousing should be possible to be able to have larger spot. Then there' a need for the software for this, too.
If ordinary pretensioned screen is used, irreversible streching when the screen is melted can be a problem. I believe that after 2 seconds your screen looks out anything but a perfect plane. I have no idea, how the screen can be planar enough when not uder a tension.
Of course it can be under a glass and pressed against a rigid plane. But the glass can absorb the laser totally and both surfaces can eat the heat in case the laser happens to be a visible wavelegth one.
Maybe you find a partner who lets you put your screen into his laser cutter. Testing reveals.
To see, how the screen behaves when melted needs no laser. Any heating, if controlled, can answer.
Assuming that the screen looses its shape as soon as it's melted, it's still possible that some at lower temperature fluid enough plastic material can clog the screen as wanted. Or laser makes the inserted material non-soluble. These both seem to be like a 2D version of 3D printer.
Etching by laser cutter probably can be used to expose the photosensitive coating on the screen. No need to be stucked to small sizes. That I also see worthy enough to be tested.