Cross-loading — this word may give a chill in the spine of climbers. It is a real terror, be it with a karabiner or knot. In the industry they adopt the simple and very straightforward solution for this. Just use super-strong metal links, a.k.a., absolute bombproof steel maillon rapides, wherever in suspect.
In climbing, whereas the same approach is indeed recommended in some cases like a group activity of abseiling or bottom-roping, the weight, bulkiness and awkwardness in use of steel maillons put off most, understandably and justifiably in many cases. Instead, climbers usually use either a climbing-rated (aluminium) karabiner or knot to bear with cross-loading. We climbers know it is not ideal, but we are also somehow confident they are strong enough for our purposes.
But how confident are you?
I have a look at this issue in this post, along with a recently created video.