All will depending whats underneath the felt. You can clean using a low moisture tool like a hydrokinetic that won't allow the water to penetrate, or even the SS uph tool as well. Low pressure, and take your time to support the fabric so that it does not stretch.
Dealing with soot you're going to want to use something with solvents in it to break it up. However, I would mix up a low strength filtration soil removal to change the charge to allow the soot particles to remove more easily, and not hurt any adhesives underneath. Make it a foam and apply the foam, and not the liquid direct with a spray bottle.
The safest bet would be using host or capture, with a brush and vacuum.