We haven’t actually tried this, but maybe something like this might help. Note that, even if this works, it is a QUICK FIX ONLY, as it is modifying the plugin file directly, and so will be overwritten whenever the plugin is re-installed or upgraded. Also, it will cause ALL sites on the multi-site to share the same list.
In bot-block.php, try changing lines in the constructor from:
$this->master_table_name = $wpdb->prefix . 'bot_block_block_list';
$this->log_table_name = $wpdb->prefix . 'bot_block_log';
to
$this->master_table_name = $wpdb->base_prefix . 'bot_block_block_list';
$this->log_table_name = $wpdb->base_prefix . 'bot_block_log';
HTH.