Just an engineer's logical thought process... absolutely zero practical experience with rear axels and differentials except for my work on my '65 Chevy Heavy Duty (C20) pick-up's axels and bearings.
The axel splined end uses slip fit (or slight press fit) into the differential U-joint.... typical and standard in most similar applications. The axel splines will therefore be forced out with relatively mild force --- i.e. the 3 lb slege approach among others.
Since it's not budging with that approach, nor with both sides of the bar being hit simultaneously (for symetric unbiased torque at the splines), it means that the splines "fit" (mating splines) are cinched up so to speak... friction is all that's holding them together unless the splines have been "twisted". If just friction, then more force is needed to overcome it... so why limit the hits with a 3 lb sledge???... gee, I use a 32 oz (2 lb) hammer just for nailing.
If twisted, plus friction, the force required is greater yet, since the sliding friction is greater... but that's all.... so more force is reqiured over a greater distance of sliding along the splines to dislodge.
Now, if you can't get the axel out by the 3 lb. sledge approach, and have to consider replacing the differential / axels anyway, then why limit the dislodging effort to a 3 lb sledge? Go for a couple of guys that can handle a 20 lb sledge each and bang on the bar (might need a heftier bar though) on both sides simultaneously 'til the axel begins to move off... then keep banging 'til it's free.
What's to lose but a few calories and perhaps some skin from a knucle or two? If you break something in the process it's still no loss as you're going to replace the differential anyway if you can't get the axel out... and even then you may find you have to replace it if you're unlucky.
The curious part is why you get engagement with high positive camber condition then no engagement with normal camber.... but that's internal to the differential.
Brute force is necessary to unstick frictional forces when those forces are high... no matter what may be causing the high frictional forces... even if somebody used super glue once upon a time on the splines... or some other adhesive, it's still a force issue. So.... more force is needed than you've been using.
Get a couple of 30 lb sledges (local tool rental joint) and have at it! What am I missing here that that 'solution' hasn't been tried before resorting to puchase of another differential?