This is in the past, so nothing you can do about it now, but if I have a schedule change that I'm not happy with but they haven't presented me with any better options, I usually close the call by saying, "Thank you for your help. I will need to research my options, and I am not accepting the schedule change at this time." It may or may not help but it at least gives me a leg to stand on if I need to make a change later.

In this case, I would try HUCA if there are flights that now meet your needs. I wouldn't even bring up the weather unless it's necessary. I would simply call up and say, "There's been a schedule change on our itinerary; we are uncomfortable with such as short connection, so would you please rebook us on X." If they push back (ie "but it's a legal connection"), then I would mention the weather angle (such as "I understand it's legal but given recent thunderstorm-related delays, I would like to have more of a buffer.").

I would think that they'd be accommodating. I recently talked Delta into making a free flight change with a 35 minute connection on an award flight (which was legal, and existed at the time of booking) without much of an issue. I find that being confident, prepared, and kind goes a long way (although it's not a slam-dunk).