If you want to get crazy buy some Kevlar Tow and epoxy it to the back side criss-crossing across the cover and on the posts lengthwise and a couple wraps around. You can use just a bit of CA glue to tack them in place. It is too brittle to use as the primary glue. But some 30 minute epoxy to give yourself more working time. A splash of denatured alcohol can thin the epoxy to allow it to flow, you want the tow surrounded by epoxy in a very thin layer. More epoxy is not stronger, it is just more weight and no added strength per se. Don't overdo the denatured alcohol as it weakens the bond and when too much is used it is severely affected. 10% is generally enough to thin and slow the cure. Use the epoxy in a warm room, 70F, or the cold can retard or prevent the proper setting of most epoxies.
New grommets and bit of silicone grease is good recommendations.
Old grommets can actually make it easier to lose a cover as they do not seat properly. Itis well worth the $10 to replace.