How much will 4mm dish weaken the rear wheel?
It depends on what you mean by 4mm of dish. When I see that it's unclear whether you mean to say that you're respacing the flanges offset from the center by 4mm each or 2mm each. (Or in other words whether you've added 2mm or 4mm to one side and taken it away from the other).

If you took a dishless hub with same size flanges, 25mm center-to-left, and 25mm center-to-right, and then added 2mm of spacer on the right and took away 2mm on the left, (changing it to 27mm center-to-left and 23mm center to right), the free spoke calculator Spocalc says the resultant bracing angles will cause the left side tension to be 85% of the right, and the bracing angles will be 4.5º and 5.3º. Doing 4mm instead makes it 73%, with a bracing angle of 4º vs 5.5º. Mid-to-high seventy percent are what disc front wheels tend to have. Mid-forties to fifty are what conventional 11-speed road 130 hubs have, as well as Campy 8/9/10.

Basically it's a pretty minor loss of strength.

I checked Spocalc's tension numbers for this against reality once a long time ago and I remember being satisfied. I am just parroting it in this case though.