First, you can't just "assume" the loan. There's about a 99% chance that mortgage contains the standard verbiage around acceleration, if the property ownership is transferred. Which means the bank could call the loan due immediately, without all the lengthy non-sense required to foreclose, if the property is transferred to you.
Second, you need to know, directly from the bank how much it would take to bring the loan current and avoid the short sell.
Now, if I'm doing my math right, even taking the high-end estimate of the rehab costs and the low-end ARV estimate, you've still got about $105k of potential profit/equity in the deal. What if you partnered with your cousin and did seller financing on the property? All you would have to bring to the table would be the $ needed to bring the loan current. If you have a contractor ready to bring money to the table for the rehab, plus moving money for your cousin, perhaps those funds could be used to cure the loan, then supplement the rehab with hard money? Lots of options, when you have that much potential profit available.
Thank you so much for the great feedback Hattie. I will definitely consider your great advice. You may have just done a wonderful thing for us. Thank you!