If I remember correctly, Oldenburg requires horses to be inspected to get C of Ps. For them it's basically saying, yes this horse is out of who owner says he is, but didn't make the cut. However for Hanoverian C of Ps it's just like it states, a certificate of pedigree!
I would think there would not be much difference between Trakehner "D" papers and Oldenburg CoP, esp. for a gelding. You might want to verify if the horse would be eligible for registry awards, but other than that, I would go with the more affordable option.