I managed to get it working through Wine. Something like this:
# Upgrade wine (https://wiki.winehq.org/Ubuntu)
wget -qO - https://dl.winehq.org/wine-builds/winehq.key | sudo apt-key add
sudo apt-add-repository "deb https://dl.winehq.org/wine-builds/ubuntu/ $(lsb_release -sc) main"
sudo apt update
sudo apt install --install-recommends winehq-stable
# Download Winetricks
chmod +x winetricks
# Install .NET 4.6
WINEARCH=win32 ./winetricks dlls dotnet46
# Install the JRE
# First, download the Windows offline java installer from https://java.com/en/download/manual.jsp
# JRE seems to only install under WinXP
WINEARCH=win32 ./winetricks winxp
WINEARCH=win32 wine jre-8u231-windows-i586.exe /s
# But other stuff only works under Win7, so switch back
WINEARCH=win32 ./winetricks win7
# Install SemanticMerge - download from https://users.semanticmerge.com/Download
WINEARCH=win32 wine SemanticMerge-188.8.131.52-installer.exe
# Run it!
wine "C:\users\$USER\Local Settings\Application Data\semanticmerge\semanticmergetool.exe"
# Once it's open, in the settings make sure to set the JRE path.
# It should be something like "c:\Program Files\Java\jre1.8.0_231\bin"