You could lash a seagrass mat on with safe string, such as undyed sisal.
In the olden days here, wheat flour glue was much used as glue for children’s crafting, esp simpler papier maché projects. A rabbit should of course not eat large amounts of wheat flour, but if you see the bun actually eating the stuff you make, you should take it away anyway because it would mean eating large quantities of paper/cardboard too and that is hazardous.
Drying time depends on the thickness of the work/ amount of glue applied, but it does take rather a long time because thesolvent is water. I have never used it to make bunny toys. You could try a smallish project to see how your bun responds under supervision.
Here’s a recipy I found on a site for artists, it’s apparently very good for making collages, better than commecially available products: 1 part wheat flour to 3-4 parts water. Whisk/stir until it mixes, then warm it while whisking, let it boil for a few minutes. take it off the heat, whisk until cool. Keeps in fridge 1-2 weeks.