Which optical bay HDD enclosure is correct for this MacBook model?

I have a 2 GHz (Early 2009) MacBook 13" (model A1181) and was looking to replace the optical drive with a hard drive from another old Mac that I have lying around.

When looking around the site, the Core 2 Duo enclosure ([연계 제품이 누락 또는 비활성화 됨: IF107-078]) specifically says that compatibility "exclud[es] Early or Mid 2009 MacBooks, MacBook Unibody and MacBook Air." The reason for this seems to be that the Early 2009 models had SATA optical drives, not PATA.

The other enclosure advertised for MacBooks, Unibody Laptop Dual Drive, says that the enclosure fits the unibody models of MacBooks/Pros but not the Early 2009 Core 2 Duo. The drive is SATA and the cable for the optical drive once it's external is SATA.

Has anyone made this install on this MacBook model and have some advice on an enclosure? I imagine that the enclosure meant for the unibody models will work, but I'm not sure of the dimensions. I have found a few off of iFixit that would likely work, but just to stay loyal to the site bringing all of these great walkthroughs, I thought I'd ask here. Thank you.