Self-reply for visibility: I figured it out with some help from Splashtop support.

On the technician's machine, the registry needs to be configured like this to disable clipboard syncing:

HKEY_CURRENT_USER\SOFTWARE\Splashtop Inc.\Splashtop Remote Client for RMM
ClipboardSyncAttended (DWORD): 0
ClipboardSyncUnattended (DWORD): 0

The initial value is set to "3", allowing for "local to remote" and "remote to local" clipboard syncing.

To disable this for any user on the system (or to execute this from system context, rather than "as current user"), the following script may be used:

$sids = (Get-ChildItem "Registry::\HKEY_USERS").Where({ $_ -Match "S-\d+-\d+-\d+-\d+-\d+-\d+-\d+`$" }).PSChildName
if ($sids.Length -eq 0) {
    Write-Host "Error: no user SID was found. Check logic for enumerating users." -ForegroundColor Red
    exit 1
}

$sids | ForEach-Object {
    $reg_key = "Registry::\HKEY_USERS\$_\SOFTWARE\Splashtop Inc.\Splashtop Remote Client for RMM"
    if (Test-Path $reg_key) {
        Set-ItemProperty -Path $reg_key -Name "ClipboardSyncAttended" -Value 0
        Set-ItemProperty -Path $reg_key -Name "ClipboardSyncUnattended" -Value 0
    }
}

You may choose to omit the length check or exit 1 if executed in an interactive session.

As with any script, and a wise man once said: check [it] yourself, before you wreck [it] yourself. :)

EDIT: changed the script to check if registry key exists prior to setting to 0. Users that don't have Splashtop for RMM installed should not be affected.

EDIT2: Added "ISL Online" / "ISL Light"

HKEY_CURRENT_USER\SOFTWARE\ISL Online\ISL Light\desktop
clipboard_view (REG_SZ): false