Step 1: Parts and Assembly
(1) 2' length of 1" Schedule 40 PVC
(2) 1" PVC Tees
(2) 1" PVC Ells
(2) 4-1/2" horizontals
(2) 1-3/4" verticals
(2) 2-3/4" pegs
Optional: A simple but very useful "jig":
a piece of scrap lumber cut to 4-3/4" width
Dry fit the lower portion (2 pegs, 2 tees, 1 horizontal) and confirm that the resulting dimension between the inner edges of the pegs is 4-3/4" (or that the center-to-center dimension is 6", a little harder to measure, but more accurate). Insert the piece on the Big Wheel to confirm proper fit. Mark the exposed pipe along the edge of fittings to serve as a stop-guide during glue-up, and confirm that your jig (if using one) produces proper spacing.
During glue-up, use your stop-guide marks and jig to prevent inserting the pipe too far into the fittings (the pipe often slides more easily and deeper into fitting once glue is applied). Work quickly, PVC glue (cement) sets up almost instantly. Have a flat surface upon which you can lay the piece during assembly to prevent "racking" (that is, to keep the Tees and pipe "flat").
Assemble the lower portion, then assemble the upper portion, verify that they line up properly, then assemble top to bottom with the vertical pipe sections.
Step 2: Optional Finishing
(Note that original seat design also has a raised curved edge near the top that produces similar bruising, depending on how your child "fits" into it -- it's worth adding padding even to the original seat!)
Here's a couple ideas:
A foam "water noodle" (a type of water play toy) works well - cut it to length, then slice lengthwise, then wrap over top bar. (You won't need to add cushioning to the verticals or lower bar as no contact is made with them when riding.)
Similarly, you may be able to find actual pipe insulation material, serves same purpose.
Or use whatever other sort of "pillow" material as you might have on hand. We had already crafted a pillow for the original seat that attached with elastic, and simply reused it here (upside-down).