The solution seems to be using pkexec with a PolicyKit configuration. This answer covers more details on the PolicyKit settings. It is geared towards executing GUI applications without requiring a password but mostly applies to non-graphical applications as well.
For my case, the .desktop file was updated to replace sudo with pkexec. Terminal= can be set to either true or false:
[Desktop Entry]
Encoding=UTF-8
Type=Application
Name=Windows 10
Comment=Switch Display to Windows Virtual Machine
Icon=/usr/share/icons/windows-icon.png
Terminal=false
Exec=pkexec /usr/bin/ddcutil -d 1 setvcp 60 0xF
Then, a PolicyKit entry had to be made at /usr/share/polkit-1/actions/com.ubuntu.pkexec.ddcutil.policy:
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE policyconfig PUBLIC
"-//freedesktop//DTD PolicyKit Policy Configuration 1.0//EN"
"http://www.freedesktop.org/standards/PolicyKit/1/policyconfig.dtd">
<policyconfig>
<action id="com.ubuntu.pkexec.ddcutil">
<defaults>
<allow_active>yes</allow_active>
</defaults>
<annotate key="org.freedesktop.policykit.exec.path">/usr/bin/ddcutil</annotate>
</action>
</policyconfig>
/var/log/auth.log seems to be where PolicyKit messages are logged if needed for debugging.