OK, I've made some changes to the task. Instead of "JD APN" I use "JD Status" to completely turn JuiceDefender off before SMS BACKUP, and then to turn it on after the backup is complete. I also put in a couple of short delays, just in case...
My task now looks like this:
1. Wait 5 seconds
2. JD Status Off: Turn off JuiceDefender. (Leave the "Status" box unchecked when setting this up.)
3. Wait 5 seconds
4. SMS Backup+
5. Wait 10 seconds
6. JD Status On: Turn on JuiceDefender
7. JD APN: Set JuiceDefender Control. (This is probably unnecessary, but it assures that JuiceDefender, and not Tasker, will be "the decider" on which data interfaces are enabled.)
When setting up this task, click on the "Tools" icon and specify the following:
1. Collision Handling: Run Both Together
2. Keep Device Awake: Yes (checked)
Seems to work reliably -- the log gets transmitted within a minute or so after phone calls and after SMS messages, through WiFi (if available) or through Mobile Data.