First of all sorry for the misunderstanding.. yes I was referring to 'config.sh'; it should build all the 'cfg' files in {z88dk}/lib/clibs starting from the templates bringing the '.lnx' name extension.
Be aware that the Ubuntu mantainer wanted to keep z88dk compliant to the common Ubuntu directory standards and had to avoid name conflicts (i.e. AFAIK z80asm is available in two flavours, the original one and the z88dk improved one).
So you should first choose what to do, either removing the z88dk packages or to 'patch' them.
The only benefit from your second choice is that you have the environment variables set.. I don't think an icon or a special menu entry exists

Let's assume you're crazy enough, it helps understanding a little more the how this compiler works.
ZCCCFG points to the 'config' files we were talking about, in a vanilla z88dk env it points to {z88dk}/lib/config but in your case it will most probably point to '/usr/share/z88dk/lib/config' or something similar, please verify it. You could compare the cfg files, and add the missing ones, then edit what you have added and hand-adjust the path comparing to an existing file, but it will make you get even more crazy, so at first just update all the '/usr/share/z88dk/lib' folder and subfolders with the new one.
Bring 'config.sh' and Makefile in '/usr/share/z88dk' and run 'make config' in that position.. it should create/update all the 'cfg' files.
Go back to your source tree.. if missing, create the {z88dk}/bin folder, then move in {z88dk}/src and make all.. reneme and overwrite the Ubuntu binaries according to what you have in /usr/bin; the target names should be:
/usr/bin/z88dk-appmake
/usr/bin/z88dk-sccz80
/usr/bin/z88dk-zcpp
/usr/bin/z88dk-copt
/usr/bin/z88dk-zcc
/usr/bin/z88dk-z80asm
Now copy the {z88dk}/include folder and subfolder to (I think) /usr/share/z88dk
Last but not least, let's rebuild the libraries.
Move in {z88dk}/libsrc and launch 'make'; if everything goes fine you should have (almost) all the main libraries in the libsrc folder; now:
mv *.lib /usr/share/z88dk/lib
This should be all.
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For the record, we should dig into the distros more often, it is a goldmine of hints for us
https://buildd.debian.org/~brlink/packages/z/z88dk.html