Is it generally safe to install VirtualBox remotely?
If you cannot afford permanently losing network connectivity, you should create an identical local testing environment with the same remote control tools and see what happens during the installation. The results should be interpret with caution: if you lose connectivity, you have a chance to try with an alternative configuration, but a successful test cannot be held as a guarantee to success on production.
With the TeamViewer's Unattended Access -feature you are likely to survive even when the installation causes the remote computer to temporarily disconnect from the Internet. That is, unless the installation prompts for something during the period the computer is disconnected. In other words, this is exactly what you should be testing.
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If the connection is not reliable with static IP address I suggested DO NOT try to install Virtual Box
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If in that PC reside a important application I suggest DO NOT try to install Virtual Box
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I suggest you manage the application with a secure SSH session instead of try install Virtual Box. This is due to as per for the screen shot the Operating System (OS) is a version of Windows.