The radioactive dating issue is unclear, totally unclear. Carbon-14 dating is considered the means of choice for dating things like religious relics. A few years ago, carbon-14 dating was performed on the Shroud, resulting in a date from medieval times. However, it now turns out that the laboratories which did the work lacked expertise in the specific area of dating textiles and may have operated with a part of the Shroud particularly prone to priestly groping and pawing over the centuries (since the Shroud got repeatedly picked up and displayed and stowed). The labs did not properly correct for the mass of organic matter—bacteria, etc—which gets deposited on an ancient textile by some centuries of priestly pawing. So it looks as though the Carbon-14 work will have to be redone. (Hey, Rome: you reading this FAQ, guys?)
There is a further source of complication with the Carbon 14. The only way people have found of simulating an image like the one on the Shroud is by irradiation from a radioactive source. Apparently, you can’t reproduce the burned-in image we see on the Shroud fibres with paint, and also cannot reproduce the shadowless, no-directional-light-source, character of that burned-in image through conventional emulsion-on-cloth photography. If the Shroud underwent something like irradiation at the first Easter (including a neutron flux, perhaps?), the post-Easter levels of Carbon 14 in the fabric could have been raised from their pre-Easter levels.
Probably it’s with the Shroud as it is with other alleged miracles, like Lourdes, in the wild and wacky Catholic world: you can with a clean scientific conscience affirm or deny.
Note, in all this, that I’m not claiming to have actual direct arguments for the proposition that Christ rose bodily. I’m arguing for it not being irrational, illogical, unscientific, to believe Christ rose bodily. My claim is only that if we can with a clean scientific conscience claim Catholicism false, we can also with a clean scientific conscience claim Catholicism true.