How can I export a full alignment from IGV as an image?
Running an IGV batch script helps increase the number of reads per image, compared with File | Save image, but it does not solve the issue by itself. 

To run a script, first make a script file like so with 1 command per line:

snapshotDirectory ./
goto chr2:28582094-28582790
maxPanelHeight 10000
snapshot chr2_28582094_28582790.png


Run the script using Tools | Run batch script menu. The importat trick is the maxPanelHeight 10000 which is what makes this include more reads than the default snapshot. 

You can also convert this into a little shell script:

#!/bin/bash
mkdir -p /tmp/igv
sleep 2
echo "snapshotDirectory /tmp/igv"
sleep 2
goto chr2:28582094-28582790
sleep 2
echo "maxPanelHeight 10000"
echo "snapshot chr2_28582094_28582790.png"


Then, just open IGV, load the relevant BAM file and run the script like this to connect to IGV via telnet (credit to Pierre Lindenbaum):

sh foo.sh |  telnet 127.0.0.1 60151