Normally adding internal longitudinal rigidizers and rings, The rings should ensure inflection points for the combined longitudinal plus tributary section of cylinder fail at much as simply supported. Then the welded longitudinal stiffeners will positively add to the compressive or bending strength.
IJR...your solution makes sense. In an axially loaded condition, it is better to keep the fix symmetric, and attaching a box section to fit inside the pipe works for that. Unless the pipe is large enough to allow a welder inside, you will need to plug weld at the points of contact.
IJR -
One thing we've done in the past is to take two steel channels, which have a depth just less than the depth of your pipe, and weld the channel flange tips to the pipe with one channel on each side. This forms a kind of ugly box with the pipe in the middle.
While it ends up not being truly symmetric, it is at least doubly symmetric. We then calculate the proportion of load that each channel takes, add to it any horizontal shear from bending forces, and calculate the stitch welds along each flange tip. We usually weld continuous for the end 24 inches or so.
You have to make sure your channel flange widths are wide enough to ensure that the channel web will not hit the pipe first, before the flange tips come into contact with the pipe.
You said "huge" pipe....perhaps there aren't channels big enough to work in this case. A series of angles could also be applied around the circumference of the pipe, forming a star pattern.
Anything similar to this would give you more area, more r, more Ix, Ix, etc. and allow for easy access/welding.
I was thinking in some underdesign, not doubling the load capacity. Even in this case the longitudinal stiffeners may work. Note that depending on the size, the plates conforming your square complement will be quite b/t slender and not much efficient; with stiffeners you may make the whole added longitudinal stiffeners area work... although the arrangement is less torsionally stout, but pipes are good at torsion.
And if you want lesser b/t and welding inside is feasible you may use a higher mumber of sides regular polygon, say, an hexagon.