Alright, I've got the solution. After fighting all afternoon yesterday with Dell support we've finally determined that there is no special Dell driver as the card is identical to the one sold by adaptec.
Despite what CS @ adaptec was saying the drivers on their website do support the Dell "version" as well as it is indeed the same card. After swapping the card through various slots on my motherboard I found that it would not work in any other PCIe slot than the one right below the graphics card which is called PCIe16_2. The Ctrl-A Menu works from any slot but Windows 10 would not recognize the card unless it was in said slot. I had already injected the adaptec 78165 driver through the INF file definition and so the card was correctly identified on the next boot.
The main chip gets insanely hot and the warning on the card saying it needed 300 LFM airflow is honestly no joke, so you really want to make sure the chip is not desoldering itself.