Your dog seems less than eager to help with this project. Fortunately, I can be of assistance. This looks like a standard folding handle for a casement window operator. I think our 39-205 will replace this; it has an 11/32" hole diameter, so make sure to check that on your own.
The only issue I see cropping up is that yours might be specifically shaped to contour to the cover. If that's the case, I'd like to see a photo of the cover itself so that we can make a better recommendation. We have a couple different parts that may meet those requirements.
Thank you for these new photos, they're a big help. The 39-107FK is definitely a similar type of nesting cover and handle, but in your case I don't think it will work since the handle is centered in the cover. Yours is offset to the side, so it won't align.
With that in mind, I think you should consider our 39-376 and 39-375. These have handles offset to the left and right and are designed to work on operators like, say, the 39-253LB or 39-239B (random examples). As long as your operators kind of look like that then I think these products will work well for your window.
Please let me know if you had any questions, and give Nacho my regards!