Slightly old info, but:
SEC filings for end of 2016 had Visa circulation at 335 million, MC at 200 million, Discover at 51.4 and Amex at 47.5.
https://wallethub.com/edu/market-share-by-credit-card-networ...
Nilson report 2016 card usage by volume has Visa at $1.549 trillion, with Amex at $0.695 trillion, Mastercard at $0.693 trillion.
https://www.valuepenguin.com/largest-credit-card-issuers
Forbes has some charts with market share info.
https://www.forbes.com/sites/greatspeculations/2017/09/07/vi...
Moreover, that still wouldn't give you competition, because then what incentive does Visa have to give you a good rate if they're the only network you could plausibly attempt exclusivity with? It's like trying to solve a lack of competition by creating a monopoly.
>contract terms in many places preventing merchants from passing on fees a
This is not a thing anymore, the card brands allow surcharging. It's prohibited by state law in something like 7 states, but those are rapidly losing court battles about it.
https://www.fool.com/investing/2017/06/20/how-did-costcos-cr...
Combine this with contract terms in many places preventing merchants from passing on fees and it gets even worse, because card networks can gain customers and increase their network size by offering rewards and then use the network effect to raise fees. If the merchant can't charge more to customers whose card network charges them higher fees, the cardholder has the incentive to use the card with the highest rewards (and thus the highest fees), so you get a ratchet of increasing processing fees despite an overall trend of falling underlying processing costs due to automation.