Imagine if the SaaS charged 100x the agreed price for the last month, refused to fix the charge, and when you issued a chargeback, they deleted all your lifetime data.
Yes? I do not understand how you would think that anything but an extremely antagonistic relationship would result from a chargeback. Not saying they "deserve to keep the money" or whatever (of course not), but a chargeback is pretty much crossing the rubicon in terms of a business relation between two parties.
Not saying "don't chargeback ever", and obviously in your scenario the SaaS is at fault, really.
Not saying "don't chargeback ever", and obviously in your scenario the SaaS is at fault, really.