I don't think you will really be able to use the Speedlite (at least the 420EX) with the "studio" flash unit.
First of all, the 420EX has no manual mode. If it did, you could put the Speedlite and the camera into manual, avoiding the Speedlite's pre-flash and trigger the studio flash unit at the same time as the Speedlite. With the Speedlite and camera in total manual exposure mode, you can then meter the light from the combination with a handheld flash meter and adjust the camera's exposure settings properly. However, since the 420EX has no manual mode, you can't do this.
If you can't get the 420EX to emit only one flash (no pre-flash, that is), you won't be able to sync the studio flash unit to the main flash from the 420EX unless you use an outboard slave such as a Wein Digital Peanut.
If the 420EX is connected to the camera and operating in auto exposure mode (ETTL), you could get the studio flash to sync properly with the main flash using the Wein Digital Peanut. However, there is an additional problem. The camera and 420EX communicate to determine the exposure settings. The 420EX emits a pre-flash which is then analysed and settings are made for the computer/flash combination. Then, the main flash occurs. The studio flash, with the Digital Peanut slave, goes off at the same time. The result is overexposure because the camera/420EX knew nothing about the existence of the studio flash.
By the way, I believe that your studio flash and the 420EX have about the same output light level.