Though both are individually weak (because of the problems mentioned), in concert, they provide a resonably compelling case.

I would have to disagree with this. I'm sure you would agree that you need to be careful before concluding that a bunch of weak evidence, put together, adds up to strong evidence. The classic example is psychic phenomena. There is lots and lots of weak evidence of psychic phenomona; including (allegedly) controlled experiments.

In the case of climate research, there is a potential problem of systemic bias. As climategate revealed, many climate scientists are more than disinterested observers; they are advocates for a position.

By analogy, imagine if Uri Geller wanted to convince the world that psychic phenomena are real. If he had 1 or 2 pieces of really strong evidence, it might be convincing. But if he presented 100 pieces of weak evidence, you would correctly dismiss his argument.