Tub cracked side lip, replace or patch up

Most tub enclosures, like the one pictured, use a 1/12 to 2" nailing flange down each side and along the top. They are also designed to be used with tubs that have an anti siphon lip along the wall parameter of the tub. It sure looks like you have very little extra relief between edge of tub/tile to the actual corner. In order for the surround to fit snugly to the wall, it must be secured using this nailing flange. I think that you may have to build out the edge of that corner, thus reducing the opening size of the pocket door by an inch or two. This would not be very difficult, but would require some extra finishing work and some changes to door trim etc. If this clearance issue is the same on the opposite side of the tub, the same procedure may be necessary. Also, if you find that your tub does not have the anti siphon lip, extra attention will be needed to seal this joint. If and when you get to the point of tearing out the tile wall, be sure to insulate the stud wall and cover it with a good 4 to 6 mil vapor barrier, tuck tape any joints and the sides and top. This will provide not only water protection, but give you some degree of sound deadening to adjacent rooms. Use only MR (moisture resistant) or paperless drywall to finish off around your new surround. A completely different approach might be to cover the existing tile with a glue on surround kit. Assuming the tile is secure, after carefully cleaning all soap residues and prepping the surface with a dewaxing agent, (isopropyl alcohol for example) trowel on adhesive and install the kit. There are actually some nice surface mount enclosures out there, usually five piece. the box stores typically only have one or two choices and are pretty cheap looking. Check out a specialty kitchen and bath supplier for a much better selection.

Lightweight spackling compound is not suitable for exterior use. Even if painted, water will effect it greatly. In addition, it is prone to cracking as it is brittle and probably will not withstand opening and closing of the door (vibrations).

Wood filler or wood glue may work to fill the crack and would be better for exterior use. The glue is probably more flexible than the filler and may be better for filling the crack. I've used the wood filler before, but usually to fill nail holes or fill in small cracks. For a few bucks and some sanding, it may be worth a go round.

Another option would be to caulk the crack. Since caulk is flexible, it will resist cracking better than the wood filler. Both caulk and wood filler would be paintable. I don't think you can stain caulk, but I believe the door is painted on both sides.

The other option I mentioned in the comments is to replace the panel/door. The panel is most likely held in by surrounding trim, so if that can be removed, just the panel could be replaced.

Another option would be to get a very thin piece of veneer wood, cut it to the dimensions of the panel, and then glue it on top of the existing panel. Since you cannot see the crack from the outside, this could be done on the inside and then painted to match.