The "big round flash" is probably a ring attachment for the flash and some manufacturers make external battery attachments for their flashes - Canon's is a CP-E4. I'm not familiar with Pentax so can't offer much help with their equipment.
Hope this gives you enough info to research for your equipment.
Just as info, I hate shooting indoor equestrian events because of the lighting. Most English events wouldn't allow flashes, but the Western events normally did. I think it's because so many English horses have a lot of thoroughbred in their bloodlines and they're a little high strung. Thankfully, I haven't had to shoot any indoor events for a few years.
BTW, here's a few pics of my granddaughter riding her Moose.
This was from her last show last year
This was when she was getting ready for the above show.
and this one is her just showing off.