If you’re sure it’s a sensitivity to Lycra then you’ll probably be best off looking for a padded liner short/underwear that doesn’t contain Lycra (or Spandex/elastane, same thing). You will probably need to look carefully at the label to be sure though!
Looks like Madison do (or did) a bamboo liner short https://www.google.co.uk/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=madison%20bamboo%20undershorts although you’d nee to see the label to be sure what makes it stretchy, as it’s usually a small percentage of elastane in stuff like this.
I have seen old cotton padded liner shorts, but from way back, not seen any recently.