How to handle it is either you take the internship, or you don't.
This is particularly true for non-programming roles which pay less. It's possible Lyft covers housing separate from salaries.
I had 4 internships, never below the median pay for my degree, and I would have certainly thought twice about doing some of them if my pay was cut in half.
And OP having that internship experience is definitely gonna help them for the future. Considering it is unlikely to get another internship (not even similar pay), OP should do it. Additionally, their half cut rate is higher than what a lot of SWE interns get too (not even considering other profession interns)>