Can this be fixed at home?

Turn the brake drum around and put it on "bell" out,, put on a coupla wheel nuts about 6 turns,, then use the drum as a slide hammer,, and tug the axle out by pushing and pulling the drum..

Depending which diff you have,, it may or may not have a "collapsable/crush spacer" which may need replacing.. If you got a parts list from the diff bill,, that may show a "spacer/sleeve"..

To be sure we need the last 3 digits off the left front snout of you diff.. 741, 742, 489, eg..

It is a 741 case with a clutch type sure grip recently installed by the shop where the fun all began. I think I recall crush sleeves as part of the kit from Randy's Ring and Pinion. Thanks for the awesome backing plate idea! I will give it a shot!