Hare and Hounds in Tetbury is nice, but will be v expensive I imagine for Badminton week. The closest one is the Compass Inn. It's a Best Western (so you can get it with Tesco vouchers if you have them!), but quite small. I've only had lunch there so not sure about the rooms but the public areas were a bit dated but clean and comfortable enough.
Cross Hands is part of 'Old English Inns'. There's normally a list of hotels/B+Bs on the Badminton website but be warned these will be expensive for that weekend at book up quickly.