According to the 1860 census Patrick Doud was living near the Post Office of Couches Gap, Tennessee. It is in Greene County, TN. There is also a Couches Cemetery located near there and there might be a remote chance of being buried there. However, like you I think if he was anywhere near his home he would have been buried there. And according to his commanding officer he did die at home. Being a recruiting officer for the Union for a spell, he may have been a target for those wearing the other color.
For a short time there almost was the State of Franklin which would have included what is now Greene County, Tennessee. It fell short by a few votes as becoming a US State. I'm wondering if somehow Eureka County may have been in the almost State of Franklin? If so, that might explain things. Again sorry for the interuption. Thx Dennis