Probably not, unless the landlord is on notice that the dog has dangerous propensities, but the SC Supreme Court has been opening the door to more landlord liability.
https://rjrlaw.com/sc-landlord-liability-for-dogbites/?doing_wp_cron=1656615015.3881790637969970703125
But renter's insurance isn't expensive. Make them buy it and add you as an additional insured, and make sure your landlord's commercial policy covers you if there's a hickup on the renter's policy.