The noise is completely normal, hat’s how a plated LSD works, there has to be some friction there. The oil reduces the friction to a moderate level, but it only really lubricates once the oil has got everywhere – it takes a while to get right in between the plates, etc. That’s why it’s tight when cold and noisier, then as it warms up and everything gets lubricated it is quieter (generally differentials lubricate on the basis of splashing – they are filled up to a certain point and as the crown gear turns around inside the casing, it splashes oil everywhere).
The noise can be the downside of having a mechanical LSD, the upside is having something that is strong and positive/effective. Helical gear / Torsen type diffs are quiet, but not as effective.
You have to use oils with special LSD additives for mechanical / clutch type LSD, as with normal gear oils the plates will really stick together very strongly and it will be very noisy.
Some oils may offer small differences in lubrication and noise, but it will all be marginal. Softer settings will reduce it too, but then the diff. will not be so effective. To be honest, Cusco LSD oils are one of the best in this respect. Cusco’s RS has lower initial torque, which will also reduce some of the noise.
There are some LSD brands that offer special coatings on LSD plates, such as KAAZ, that are quieter. But generally the noise just means it is working – later in life when the plates become worn they will not be so noisy, but the diff. will not be so effective either.