I am pretty sure that HSS (high speed sync) is special feature of strobe, doesn't come for free with any light source. HSS requires that flash fires multiple times during shorter exposure, so it requires special bulbs and larger capacitors that can hold and release quickly enough charge for several flashes per exposure.
Maybe you can live with light sources providing constant light output? Those can be combined with HSS as far as I see.
Using HSS strobe with non-HSS strobe and exposing at high speed (let's say 1/1000) will cause funny effects. If you use HSS for main light and have regular strobe in the background, model face will be properly illuminated, but background will have only stripe highlighted, as shutter movement defines when light gets on the sensor.
Finally, you didn't specify what shutter you are using. Leaf shutters have higher sync speed, so HSS or no HSS, you can use greater shutter speed. Curtain shutters (default in 35mm DSLRs) require HSS. Also special case of electronic shutters (basically when sensor controls exposure time) allows whole sensor area to be "active" at the same moment, ready for strobe pulse.