I'm still unconvinced.
"Feral rats. Our sample consisted of 6115 (2886 males, 3229 females) wild Norway rats (Rattus norvegicus) that were captured in the central alleys of high-density residential neighbourhoods using single-capture live traps, while rural rat populations were sampled from parklands and agricultural areas in areas surrounding the city [12,13], between the years 1948 and 2006."
I can't cross-check the methodological validity since neither articles are available online (the second one appears to be, but if you follow the links you end up with a n unrelated 1975 article on beach voles :-). No info is given on the distance between the traps and human habitations.
Also, no idea on how comparable the sample from 1949 is with the one from 1989. They were not written by the same authors, and the 1989 authors don't cite the 1949 paper (https://scholar.google.com/scholar?cites=1484756086713821658...).
Had they compared rodent populations from places that the obesity epidemics has yet to reach (read: poor countries), it would be more convincing.
For marmots, the proposed hypothesis was that the longer summers let them eat more food and shortens the hibernation period during which they usually lose weight.
If you want to know whether atmospheric CO2 has an impact on rodent weight, you can devise a controlled experiment where they live in, and are fed with plants grown in controlled atmospheres. No idea if such a study has been performed.