No need. It mutates itself. The propagation of genetic strains of Covid-19 can be viewed here:

https://nextstrain.org/ncov/global

Our ability to do this for every single Covid-19 case in the universe is, I imagine, limited only by funding/infrastructure/training.

We basically already know how viral viruses are because they undergo mutations, and we can trace which strain people have and where they caught it from by sequencing their DNA (technically RNA) and looking for the virus's "ancestors".

See for example https://www.eurekalert.org/pub_releases/2020-10/du-gt100120....

Also you wouldn't mutate a virus with C14 or similar radiactive isotope, because that can't be encoded in its genetics, and such a virus would use any old Carbon atom (most likely normal C12) to replicate.

Finally, mutating a virus and releasing it seems ethically dubious in the middle of a pandemic...

Where would it get the carbon-14 from? Virus particles (virions) are made by and from their host. So you'd need a human made from carbon-14 first.

AFAIK there aren’t methods to do that in vivo, unfortunately.