read bash environment variable

You need to source your script instead of running it.

. set-vars-script.sh

You can not use braces in form ${VAR[ARITHM_EXPR]}. But you can always extract value of these variables to variables with VALID names:

  $ set | sed -n '/CLIENT\[user-agent]=/{s|.*=||;p;q;}'

Why you don't use Perl/Python for scripting? This resolve your problem:

  import os
  print(os.environ['CLIENT[user-agent]'])

Probably you should declare the array CLIENT before assigning to its elements:

declare -A CLIENT

Can you edit the script assigning all the values to the variables marked for export to subshells? Can you create a temporary file. Bash can't export arrays, associative or indexed; but you can use the BASH_ENV variable to name a set-up file for non-interactive subshells. Be sure to use declare -A CLIENT before assigning values. Then add something like this after all assignments to the CLIENT array

    declare -A -p CLIENT > /tmp/bash_env_tmp
    export BASH_ENV=/tmp/bash_env_tmp

A new non-interactive subshell will read and execute the file before processing its commands. If the shell is invoked with the -c option, then you can use a command substitution with on the line calling the subshell to supply the output of the same declare operation as the first commands in the new shell.

Direct answer: bash doesn't consider CLIENT[if-modified-since] a valid variable name, so you need to use something else to get its value. Fortunately, printenv can do the job:

printenv 'CLIENT[user-agent]' >> "${LOG}"

Note that this doesn't use the ${...}; that's bash's variable syntax. Also, it's a really good idea to quote the name (either single- or double-quotes would work), since square brackets are filename wildcard characters, and using those without proper quotes can lead to weird problems.

If you want to pass the value as an argument to another command, use printenv in a command substitution:

someCommand "$(printenv 'CLIENT[user-agent]')"

Note that you should almost always double-quote variable references, command substitutions, etc. I recommend running your scripts through shellcheck, and following its recommendations to improve your scripts in this, and a variety of other ways.

Another option is to copy the value to a bash variable and use that:

CLIENT_user_agent="$(printenv 'CLIENT[user-agent]')"
someCommand "$CLIENT_user_agent"

Explanation and clarification: I'm seeing a lot of confusion in the comments and other answers about what's going on here. The source of the confusion is that in bash syntax, CLIENT[user-agent] (used in a variable-name-like context) would refer to an array named CLIENT, indexed by user-agent (which itself might be either a string or a math expression, user minus agent, depending on whether CLIENT is an associative array or numeric-indexed array).

But, at least if I understand the situation correctly, the [user-agent] is just part of the environment variable name itself. At least in Linux, environment variable names can contain pretty much any characters except "=".

bash, on the other hand, only allows letters, numbers, and underscores in names, so it cannot (directly) set or access environment variables with names containing other characters.

Also, environment variables cannot be arrays, only plain strings. Arrays are a bash (and ksh and zsh and...) thing, not something the operating system's environment supports.

The notation:

 CLIENT[text]=Othertext

is used to set an index (text) in an associative array (CLIENT[]), and you can only do that in bash after you have used declare to tell the shell that the array is associative. Without that, an array is normal, and can only have integer indices.

So, you say you have a program which "gives out" environment variables. I'm assuming the variables are stored in a file, so you might:

declare -A CLIENT
source /path/to/variablefile
declare -p CLIENT
# or
echo ${CLIENT[user-agent]}

The first declare makes the associative array. You then source the variable file to do the assignments in the current shell rather than a subshell which will be lost when it finishes. Everything else is for demonstration purposes.

UPDATE

Though, come to think of it, this won't quite work as-is. Any line that tries to assign content but has a space in it will break. So:

CLIENT[if-modified-since]=Thu, 02 Feb 2012 12:09:40 GMT

is illegal. You can assign the variable correctly if you quote the data, of course. But these variables are created by a program, you said, and we're not here to debug that program. So,

declare -A CLIENT
sed -i 's/\([^=]*\)=\(.*\)/\1="\2"/' /path/to/variablefile
source /path/to/variablefile

The sed script simply quotes the variable assignments in the file. This seems to work on the file you provided. Quotes are good.