I wasn't aware that you couldn't restore from a backup if you are using a trial version. What is the exact message you are getting when trying to restore?
You can manually extract files from the backup zip files using a program such as WinZip. Any large files that OneCare backup split into parts can be rejoined with this utility:
http://www.freebyte.com/hjsplit/
I do suggest contact support, though, as I'm not sure that there is a limitation for the trial in restoring from a backup made while a subscriber.
An alternative may be to go ahead and subscribe and the cancel the subscription as soon as you've completed your restore and removed OneCare from the new PC.
You'd need to contact support to cancel and get the refund.
How to reach support (FAQ) - http://forums.microsoft.com/WindowsOneCare/ShowPost.aspx?PostID=2421771&SiteID=2
-steve