What political advantage, exactly, did Clinton see in championing climate change legislation? Because, again, prior to 1997, there was virtually no political divide on climate issues.
It wasn't any kind of domestic politics, it was international politics. Long before he signed it, he had already stated that he would sign it, as a first step to keep the discussions and the progress moving forward, especially in light of the fact that it was free market based and not cap and trade based (which is eventually became, anyway), but that he wouldn't present it to the Senate for ratification unless and until the developing nations are under the same goals as the industrialized nations.
The Byrd-Hagel Resolution was voted on in July of 1997, and the Kyoto Protocol meetings didn't happen until December of 1997, where the broad outlines of the protocols were agreed upon. The Byrd-Hagel Resolution wasn't about partisan politics, it was about disapproving of any international agreement that (A) did not require developing countries to make emission reductions (as Clinton already stated) and (B) "would seriously harm the economy of the United States" because the treaty would exempt, in one way or another, about 80% of the world's population, including China and India. Clinton refused to submit the treaty to the Senate, as did Bush, and thus far Obama has also refused, all for the same reasons.
But the fact that Clinton signed it was all the Republicans needed to take full political advantage. They completely dismissed his comments and reservations about the Protocol, such as it's not what the US wants, and that it's only a first step. They completely ignored Gore's comments of, "As we said from the very beginning, we will not submit this agreement for ratification until key developing nations participate in this effort. This is a global problem that will require a global solution." All that mattered to the Republicans was that Clinton signed it, dismissing totally that he did so as leverage against the EU, Russia, China and India. For the Republicans, it was domestic politics, one hundred percent.
Remember, things between the Clinton left and the Republican right were already rather polarized, and this all took place in the midst of the Paula Jones lawsuit, where the Supreme Court ruled in May of 1997 against Clinton that her lawsuit against a sitting president could proceed, and immediately after the Kyoto meetings is when the Lewinski scandal broke. After that, whatever Clinton wanted, the right wanted the opposite, and a stance on climate change quickly became incorporated into the agenda of the right.