A weight does NOT necessarily raise the outer edge of the record. A properly designed reflex clamp is a well-known engineering solution to this problem.
I use a very thin Delrin shim of about 75 mm diameter and a 2+Kg steel weight with a Delrin ring around the bottom. The result is a shim that forces the label up and a weight around the edge of the label which forces everything else down.
Benefits are improved flatness, obviously, and also an extremely good marriage between record and the 1" graphite mat, resulting in increased clarity and dynamics.
This approach is not suitable unless the thrust bearing of your turntable is sufficient for the task. My air bearing is..