Favoring those who are price sensitive isn't necessarily benefitting those who are less wealthy (this wasn't explicitly stated, but it's often implied in praises of price discrimination, please let me know of this isn't what you meant with this comment). Wealthier people often have time to find better prices, or have others find better prices for them. Also, people with better information skills are often better at identifying the best price (and people with such skills tend to be better off in general). A poor person balancing several jobs may not have much time to research prices.

On priniciple I'm usually okay with rewarding diligence and thoughtfulness, but penalising those who lack the ability to perform this diligence and thoughtfulness may not be the best thing in this case.