Very neat! I did some digging around before posting, it's a very interesting point in showjumping history for sure.
Given the time in history and the fact that this horse was bought and owned in the US, it is highly unlikely that he had any warmblood in him. Most warmblood breeds were not seen outside of Europe until well after WWII - most were not imported into North America until the late 50s-60s.
American horses from that era had a lot more "harness horse" blood in them than you would think. Jenny Camp was a notable U.S. Olympic horse at the time, and she was mostly TB but was thought to have some standardbred blood. That's definitely possible for this guy; saddlebred, cleveland bay, hackney, arabian or morgan blood is possible as well. Reading about the US Army Remount Service might help you get an idea of the horses your great grandfather may have worked with at the time - not sure if this one was a product of it since he was purchased at auction, but many of the horses he went to the Olympics with were.
I wouldn't rule out full TB, they were a dominant riding horse breed and some of the non-racing (ETA scratch that, turns out Gunrock in the link below did race!) studs were fairly coarse in appearance. Look at this guy (full TB): Home
Thanks for sharing that bit of family history with us!