The RX10's built-in flash is TTL-only. With TTL, the camera tells the flash to send out a "preflash" burst of light that it can meter, and then the camera automatically adjusts the flash's power output to get what its auto-exposure system thinks is good exposure, before sending out the main flash burst while opening the shutter for the exposure. This pre-flash is setting off your studio flash's optical slaves too early.
The easiest solution would be to use an external flash on the hotshoe of the RX10 that can be put into Manual mode to eliminate any preflashes, but yes, you could also simply use the hotshoe of the RX-10 with radio triggers to set off your Portaflash heads. The 336VM has what looks to be a 1/8" (3.5mm) miniphone sync port. You'd just need the right cables to connect a receiver to each light.
You do have to be careful not to pick a transceiver unit that requires a Canon/Nikon TTL signal to switch between Rx/Tx mode, but a lot of cheap radio triggers are liable to work (e.g., Yongnuo RF-603II, Phottix Strato, RadioPopper JrX, etc. etc.) Radio triggers these days can be relatively cheap and eliminate the line-of-sight requirements of optical slaves, so they're pretty popular.