I sounds like a nice control theory problem to see if you can stabilize an orbit near L1 with a solar sail and a minimal gyro, while fullfilling the role of blocking radiation maximally; I'm not very qualified though. I think you'd need to place the object not actually at L1 but at an orbit closed to the sun so that mean reflective impulse at the sun-veanus direction generated by the sail cancels out the extra gravity.

I love this kind of proposal because it's hard to tell off the top of your head whether it's absurd or completely plausible, given some technological level. You have to look at the basic numbers.