As you can imagine I am somewhat curious about a "Hospital Log Book". I am sure such records were kept that was the military way of things. Virginia Hospitals were the best documented in records. But the farther west you go the more scarce any records become. Yet there are tons of civil War records still held in the War department collection in the National Archives which have never been examined to this day for their historic value.
Most of the medical information that I have seen for Arkansas Confederates comes from 1st Sargeants reports and Hospital reciept contained in the individual soldiers combine service records. Again in Francis' case are we talking about him being in a Union Hospital in Fort Smith, or a Confederate Hospital in Little Rock? There was a difference in the records keeping of hospitals on different sides. I will bet that you were looking at a Union Hospital Log Book for the Fort Smith hospital in 1864.
As for the pension application it was possible for a person to obtain a pension on the sworn testamony of two persons who served with the person making the application. I am not sure why Millie could not have recieved such a pension on Benton's service by using the testamony of Francis and Alvin.
However, pensions for CONFEDERATE service were not authorized until the 1890's. Again when you are talking about Millie appling for a pension for Anderson's service that pension was for service in the UNION army which was authorized much earlier than the Confederate Pensions which were paid by the individual state and not the Government. So as with the Hospital records you are comparing two different things here. Union Records and procedures vs Confederate records and procedures