Seems the download and update process works fine here - I tried it several times.
When does the download process lock up exactly? Does it always lock up in the same time? Does the whole emulation lock up (means you cannot even reboot) or AmiKit only (you can reboot), or just the LiveUpdate program?
What about firewall on PC side? When LiveUpdate locks again, try to switch to Windows. Maybe there's the firewall requester that prevents LiveUpdate from running correctly.
If I understood it correctly, you managed to download the whole update once - without the lock up. But the update was not installed after you rebooted. Then please check the AmiKit:Expansion/RecentAction.log and tell me what text it contains. Sometimes it can happen that if you reboot too quickly, the recently modified files are not stored on your HDD. That's the reason why eg. MorpheuZ and other programs wait for 3 seconds before the reboot requester appears, you know.