Your technical manual is HERE.
http://h10032.www1.hp.com/ctg/Manual/c03612798
You can look on page 172 to see the cards HP certified at that time for the SFF Elite 8300. You certainly can do better than those, from 2012.
That SFF has only a 240W power supply. HP has been sticking with the ATX standard for the PCIe x16 slot of it being able to provide 75W up to the card, but that is not the whole picture..... you have a limit on the total 12V power from such a small power supply. That will be on the power supply label.
We use the Quadro K600 or K620 cards in these without issue. Per nVidia directly those use 41 vs 45W max TDP, and we don't see any better performance from the K620... it just has 2GB instead of 1GB DDR3 and draws a bit more power.
Sounds like you'd be trying to put 10 pounds of flour in a 5 pound bag with the 75W 1050 Ti..... maybe will be fine with the 60W 750 Ti. These are not gaming boxes. Turn off on-processor graphics. Use SSD(s) instead of HDD(s).
There's lots of discussion about wattages of power supplies and max TDP of video cards for you to research via google. Good news is that the newer cards draw significantly less power for same performance. So, start as new as possible and work back. Don't even think about cost....... go for it and let us know.
