Amy Vanderbilt wrote books on etiquette and published a newspaper column to answer questions. The rules were a bit different then. For example, being "pinned" was somewhere between "going steady" and being "engaged," but it was okay to date other people? And what about a business lunch -is it okay for a woman to pick up the check when she's dining with her firm's client?
“Yes you may, saying something such as ‘This is business—you’re the firm’s guest.’ If the bill is to be paid at the desk, quietly put money to cover it on the check and ask your customer to take care of it. Either leave the tip yourself or ask him to take care of it out of the change. Try to avoid passing any money yourself, for other diners in the restaurant would not necessarily understand the circumstances.”‘
So it's okay for a woman to pay, as long as no one sees her. Gotta protect the man's dignity, after all. Read more at Death and Taxes. Link
It was more than that. Women dining alone were considered to be prostitutes and were often turned away from restaurants. (Two or more women at a table was tolerated.) A man and woman could also have the woman's virtue brought into question back then with money changing hands in public places decades ago. In order not to cast suspicion on either party's integrity, a woman paying the bill was done in a round about way.
The man's 'dignity' was a lesser consideration but still a valid observation.