I would stay away from round rails, even mounted ones. The round carriages just don't have the minimal backlash that profile rails do. Something like these:
https://www.amazon.com/ReliaBot-Linear-Carriage-Printer-Machine/dp/B07B4DWWZC/ref=sr_1_1_sspa?crid=2MZBCRDI7FCWP&dib=eyJ2IjoiMSJ9.Llw_TKV7mmhxPx67bYg1ZJfyMppRTvf1VZfWN-_dBrQY6Ol53Z4t7Y3fnYOKj7gI63L6wxYgn30m60q-e3tdQwy8_RorDkDpywOceuJsg1s2GPvNZNKnfnPjB3nA7Udt88v7OHhY7qemDcP7ZYt3ybuwPqnayOvcBz-GdEWgU6dkf38Jx9zpBse0ZPKFdQ6acvRRc6cm5KcC7F83cx7s8N4nivSX-y_ujWGi0dzv14vuh9poOFOX0gLNfb71mDFFMrTNs1XJnB71i1znMtc-bsIiPNDvXj1ZS43F75Rzzyg.N2NFCOlkzYK-hEGLNffnZvPJH8xaAa152ChVYNNSLDM&dib_tag=se&keywords=size+12+linear+rail&qid=1750385121&s=industrial&sprefix=size+12+linear+rail,industrial,127&sr=1-1-spons&sp_csd=d2lkZ2V0TmFtZT1zcF9hdGY&th=1
If you do design for these, it is suggested to have a reference edge on one side that you can push the rail to. Tighten down that rail, then 'float' the other rail in using the components that tie the two together. You can get wide rail that has a decent amount of rigidity about the x-axis of the rail, but having two spread as far apart as possible will add to the stiffness.
I did not intend to say that the rails are the only issue, apologies for any confusion. The rails may add to the problem.
If you decide to tweak any of your designs, you can easily model in space to add metal angle or bars to add rigidity. You could also the basic outer shape as a thin-ish 3d print and fill the voids with epoxy mixed with tile grout, which will give great rigidity and damping.
Finally, if you have large areas that would use an infill, you would be better off creating an internal matrix of structure aligned to counteract your predicted bending forces. Look at how airplane wings are built, they have spars the entire length of the wing and ribs perpendicular to the wing box. Bridges are the same.
PLA is a very stiff material and very good in compression. Adding aluminum plate or angle, bolted and epoxied to the outer sides of the the item will give a lot of tensile strength (CF would also work).
Feel free to PM me if you want to discuss more in private.