As for the LDS response and its effect on missionary work in the Southern States I can only find this notice from the Liahona.
October 26 1918. General missionary work, such as holding meetings and tracting, has been suspended during the epidemic of influenza. The missionaries, however, are visiting the sick, the needy, etc. and offering prayers to the Lord in behalf of the sick. Several of the Elders of the Southern States Mission are ill with the flu, but they are recovering rapidly. Liahona 16:1166
Interesting since there were still some meetings held, but the reporting was more subdued. Where in earlier reports in 1918, there would be names of missionaries and where they were assigned. In October there was nothing. In the months that the epidemic was beginning to let up, the reports only noted that a meeting was held but no names were listed. On the 17th of November a meeting of the East Tennessee Conference was held at Chattanooga, and on the 16th of December a meeting of the Middle Tennessee Conference was held in Nashville, but were quite subdued. The regular description touting the progress of the missionary effort was replaced with a perfunctory statement of fact. As for the missionaries themselves, none of the missionaries serving in Tennessee were sent home due to their illness, though it was likely many were sick.
More interesting was the change in missionaries being called. Quietly, the church stopped sending missionaries to the Southern States. The number of missionaries was already low because of the war, but between October 14th 1918 and February 24 1919 no missionaries were sent to the Southern States Mission; a gap of three and a half months. By then the epidemic had subsided.