Nevermind!!
A little experimentation was all it took.
kinjiru:/etc/java # ls
. .. java2.conf java2-jre.conf README
kinjiru:/etc/java # cat java2-jre.conf | less
# Configuration for java2-jre package
Priority: 40
Vendor : Sun
Version : 1.4.2
Devel : False
JAVA_BINDIR = /usr/lib/SunJava2-1.4/jre/bin
JAVA_ROOT = /usr/lib/SunJava2-1.4
JAVA_HOME = /usr/lib/SunJava2-1.4/jre
JRE_HOME = /usr/lib/SunJava2-1.4/jre
JDK_HOME =
SDK_HOME =
JAVA_LINK = SunJava2-1.4
...etc, etc.
# export PATH=/usr/lib/SunJava2-1.4/jre:$PATH
Ja! The correct executable was located at JAVA_HOME. LimeWire works perfect now.
I initially tried to export JAVA_BINDIR (= binary directory?), but got a load of wierd error messages. I had to, as root,
# xauth merge /home/my_home/.Xauthority
# xhost + localhost
Then the installer ran, and viola.