There is no work around for license keys. Each i5OS licensed program product and the OS itself are keyed based upon the serial number. IBM gives you 70 days on a restore before it will not work any more. The thing to do is restore everything and then take the license keys for the 870 and install them on that new recovered LPAR. These are the correct IBM license keys to use.
For third party applications, you need to contact each vendor and get temporary license keys or buy a DR license if you are always going to restore the 820 to an LPAR on the 870. Most companies will give you a temporary key good for up to 30 days. This also greatly varies based upon vendor.