Misty Fuse would work well for this. It is extremely lightweight and does not add bulk.
https://www.amazon.com/Attached-Mist...dp/B005H2050Q/
Edit: I noticed the heading on that link said "interfacing", but if it's the version of Misty Fuse that I use, it is not interfacing; it is a fusible web, so it fuses both sides. You might want to double check that, as an interfacing that fuses only one side won't work for your needs. However, the package looks like mine, so I am thinking it just a mistake in the description.
A couple of tips for using Misty Fuse. Use parchment paper on top of your ironing board, and another sheet of parchment paper on top of what you are fusing. This protects both your ironing board and your iron from getting fusible on them by accident. Also, don't work if the Misty Fuse has some wrinkles in it. It is so thin and fine, you can fuse several layers of it together without adding bulk.