Did you try to run a check and repair of the backup database through the console? The backup database is sensitive against any kind of impact, not very robust and not redundant. So it may fail, and reading such a failed segment may cause the laptop to reboot.
What happens if you try to open the laptop backup from the stationary PC?
If this runs successfully you may have run into an incompatibility of drivers or software (virus scanner etc) on your laptop.
Best greetings from Germany
Olaf