To expand a bit on the answer of NMech:
I would recommend a pulsed laser, as it allows a way more controlled energy input into the fabric. Some information on how to pick a pulsed laser for a process. A rule of thumb when developing a process with a pulsed laser is:
The pulse makes the process, the power makes the throughput.
What does this mean? When you pick a pulsed laser, they are usually advertised with their average power, which will on top of that be way lower than what you are used to from cw-lasers. While a 1 kW cw-laser is nowadays pretty meager, a 1 kW pulsed laser is already quite the beast.
For a pulsed laser, the way more important value is the maximum pulse energy that hits the material from a single pulse. This value plays a big role on how the light interacts with the material and needs to be carefully picked when developing the process (a good laser should have many different pulseforms with different max pulse energy and pulse energy distribution over the pulse duration, giving you a lot of choices to work with). The average power only decides how often these pulses get fired.
Example: You found a vendor that offers two similar pulsed lasers, one with average power of 50 W and one with 100 W. Which one do you take? The 100 W will fire the exact same pulses as the 50 W, they won't have twice the energy. It will just fire it twice as often and thus allow you to process more material in the same time.
All in all, don't be put off when you see a pulsed laser with seemingly low average power, it might do the job just as well, just slower. Instead, look at the pulse energy. Hope this helps a bit with picking a laser source.