Good question. Republicans in particular need to worry about primary challenges in their science-illiterate districts, and many may be wary of attaching their name to this directly. But there are a good handful of deep red congresspeople who are happy to discuss policy details around carbon pricing, even if they can't politically afford cosponsorship [0]. I imagine that several of those may be willing to vote for such a bill. This is why it's particularly important to write to conservative lawmakers – most of them understand that this makes economic sense, but citizens have to demonstrate to them that they won't lose votes over this kind of legislation.
Also, consider that if Democrats gain control of the Senate in 2020, they can pass this easily, and once such a fee-and-dividend scheme is around for a few years, it will likely become very popular with citizens (because of the dividends [1]) and big business (because of the predictability, compared to regulations/bans/subsidies/political favoritism [2]), so it won't be easy to get rid of it again. Instead, the battleground will shift, as we currently see in Canada, to "how fast do we increase the price per ton?" and "which industries get special treatment?".
[0] https://citizensclimatelobby.org/climate-solutions-caucus/
[1] https://citizensclimatelobby.org/household-impact-study/