I think your best bet would be to exchange the backers before you cut the graphics.
Set up your laminator with the branded liner in place of the laminate.
The rollers would set the new liner better than anything you could do by hand.
Crank a roll through and then plot out the graphics.
If you have to cut first and then try to change out the liner you might try rigging up something similar to this:
http://www.speedpress.com/products/2/44
Get low tack clear transfer tape and stretch it out on some aluminum extrusion with a couple of hinges to hold it onto the table.
If you could get the transfer liner low-tack enough to let the decals go onto the new backer it may work.
When we used to set a lot of multi colored/layered vinyl up we didn't want too much grab from our mask paper we would condition it by rubbing it down with a cloth so that it was not too sticky.
You need enough grip to get the old liner off but not too much to keep the decals from releasing onto the branded liner.
If you wanted to go high-tech maybe a really strong vacuum table......
wayne k
guam usa