The tough thick strands are fine for buns to eat. In fact you want a hay with both leaf and stalks. The leaf provides most of the nutrients, the stalks provide indigestible fiber to help with good digestive motility. The mature stalks are also good for tooth wear. So your bun eating the stalks is a good thing, as long as he is also eating some of the leafy strands and is maintaining a healthy weight.
Some straw would be fine, but straw has very limited nutritional value and is mostly indigestible fiber, so you would also need to ensure your bun is getting enough protein and nutrients from other foods if you decided to give him some straw.