I'm not sure how cheap you're trying to go, but I have a cheap GoPro wannabe camera, and have used it to film tuft testing on my car. The image quality was good enough for that. The audio quality is terrible, but you don't need that for tuft testing.
Anyway, the camera was around $70. And where GoPro wannabes are concerned, that's pretty cheap. It has a mounting bracket with a 1/4'-20 mounting lug (so you can attach it to various camera mounts), a view screen, and rechargeable batteries. You can also control it via wifi with your smartphone. And it comes with an assortment of camera mounts.
Here's an Amazon link --
https://www.amazon.com/AKASO-Adjusta...dp/B07JR1XZ78/
In the video below, the clip starting at about 4:16 was filmed with the Akaso camera on a suction-cup mount. (I took two video clips and squashed them to fit on the screen at the same time)
That said, if this is just going to be a one-time use, that's probably overkill. But you might find other uses for the camera.