Not heard about a workaround, but repeatedly cancelling the sign in prompt didn't work for me, it eventually took me to a "Sorry but what the fuck is going on?" page and told me to quit the software.
No, Lovefilm doesn't work without being signed in to PSN. Why is anyone's guess.
I thought someone had found a workaround.
Can't you just keep cancelling the sign-in attempt until the Lovefilm software gives up and goes home (to Lovefilm, taking you with it)?