Are cattle really contributing to global warming?
Let's just talk about cattle here and ignore other factors related to cattle production for now.
The current cattle number (93 million) you found is probably the total number of dairy cow and beef cattle. In your argument, it is possible that bison has much more emission than cattle on pasture. But the emission from cattle in feedlot and dairy farm is much larger than the cattle on pasture because of their diet chosen by us (human).
Typical beef cattle (steers and heifers) in the U.S. spend one year on pasture, and 5-6 month in feedlot before slaughter (unlike Argentina beef cattle which are fully raised on pasture). In feedlot, these young cattle (yearlings) are intensively fed with high protein feed in order to get them fatten up in a very short time. The U.S. dairy cows spent most of their time on dairy farm (unlike the Swiss cows who are raised on pasture). They live for about 5 years and are continuously fed with high protein forage to ensure their milk production. Both beef cattle and dairy cows have a high protein diet that result in high emission of methane (the impact of methane is 25 times worse than CO2 in terms of warming).
The closest comparable cattle to bison is beef cow which spend most of its life time (about 10 years) on pasture and are only occasionally fed with low quality hay when there is a shortage on pasture. But again, the life span of a beef cow may be different from the buffalo and beef cows may eat more because they are basically pregnant very year(!).
California has this new policy that tries to cope with dairy cow methane emission from their manure. If you are interested, there is a recent study done by University of California Agricultural Issue Center here (https://aic.ucdavis.edu/wp-content/uploads/2019/01/DairyManure-1.pdf).