It is possible to use Livestream Studio 6 as a software switcher, and use LiveU Solo as the transmission device for the program output of that switcher.
The basic setup here will be to route the Program Output of the Studio 6 switcher over a baseband output from your laptop (HDMI or SDI) to the HDMI or SDI input on your LiveU Solo, just as if you were plugging in a camera.
Unfortunately, Livestream Studio 6 will not recognize your built-in HDMI port on the laptop, or an HDMI port added by a general dongle (such as the dongles available from Apple themselves), as an "output device". It does, however, recognize broadcast output devices such as those from Blackmagic Design. See the example below, using the Blackmagic Design UltraStudio Monitor 3G device, which attaches to your laptop over USB-C. So you will require one of these supported output devices, even if your laptop already has an HDMI output.
The first step is to fully install your device, such as the UltraStudio Monitor, to your laptop OS. For the Blackmagic Design devices, that means downloading and installing the Blackmagic Design Desktop Video software to your laptop. This may also want to install a new firmware onto your device itself.
Once the software recognizes the device is present, you are ready for the next step. Here is an example of the Blackmagic Design Desktop Video software recognizing a UltraStudio Monitor plugged into a Mac laptop.
Studio 6 Software
Once those steps are done, in Studio 6, open the Settings panel, and you need to change two settings:
In Project Format
Select the format you want to output over SDI or HDMI, note you cannot use "Optimize for Streaming" with this setup.
Once set to the format you want, go to the Outputs tab and in the top section, of Outputs, click the plus sign to add a new one and from the Device dropdown, select your device and its output interface you want to use (HDMI vs SDI).
Once complete click Save for both settings.
Using The Setup
Once setup this way, the plugged in device and its baseband output (again, HDMI or SDI) will always output the program output of the switcher - that is, even without ever pressing "Go Live" in Studio 6, which would activate Studio 6's own ability to send a streaming protocol such as RMTP.
In this way, you can think of it almost like your laptop is now a hardware switcher, and you are plugging the program output of that traditional switcher into your LiveU Solo.
Plug in the HDMI or SDI of the attached device to your LiveU Solo, and when you want to transmit, start streaming on the LiveU Solo - and the LiveU Solo only. Again, you do not need to use "Go Live" in Studio 6 - in this setup, the baseband output coming from your attached device (like the UltraStudio Monitor) is the output of your switcher, and the LiveU Solo is your only transmission device.
You can see in this example, that while the Studio 6 interface still shows the Switcher interface, the output coming from the UltraStudio Monitor is the program output of the software, without any desktop or UX present.