I have an update: everything works fine when started via systemd when I change the unit file to Type=forking. Wondering how reproducible the problem is on other platforms...
# /etc/systemd/system/backuppc.service
[Unit]
Description=BackupPC server
After=syslog.target local-fs.target
[Service]
Type=forking
User=backuppc
Group=backup
ExecStart=/usr/local/BackupPC/bin/BackupPC -d
RuntimeDirectory=BackupPC
PIDFile=/var/run/BackupPC/BackupPC.pid
KillSignal=SIGTERM
Restart=always
RestartSec=1
[Install]
WantedBy=multi-user.target