Simisteven:
If you work on ladders, you should wear the correct foot ware, as mentioned above by NickE, flip-flops don’t cut it, even on sloppy Fridays. Fine em, for improper foot wear, they’ll have something else to think about that way.
Flat bars as rungs tend to be (can be) more slippery than most round bars. Both should have some anti-slip, anti-skid, treatment or ridges (deformations) formed in them.
Some safety codes and industry codes do not allow welding to, or on, any safety appliances and ladder rungs and the like. The AAR and FRA for example. This is because of the potential that welding imperfections, arc strikes, and the like can cause cracking and hard spots in the rung or piece, which might cause sudden failure of the part when loaded. The thinking goes that, a bolted, riveted connection might loosen and would be noticed for repair, but not be as likely to fail suddenly, as a weld might.
You can buy steel bar stock (3/16" x 1.5" for example) which has slightly rounded edges due to the rolling process. There are also bar sized channels (approx. 1.5" x .5" legs x 1/8", other sizes too) which might work with the legs down. If I were going to try to do this, I would cut the bar to length to fit btwn. the styles and weld them at their ends only, to the styles. I would do a little testing and form these bars/channels so they had a 3/16" +/- circular concave upward. Thus, when you applied them, you would weld one end to its style, then press the new rung down to weld the other end. This bending should prestress the two rung pieces together and prevent pinching. Don’t weld out in the middle of the primary rung. Weld process and material compatibility for welding is all important. Check with your local safety guys and OSHA.