Just VAR=value defines a shell variable. Environment variables live in a separate area of process memory that is preserved when another process is started, but are otherwise indistinguishable from shell variables.
To promote a variable to an environment variable, you must export it.
Example:
VAR=value
export VAR
or
export VAR=value
If you put the above into .bashrc, the above value of $VAR should be available in the script, but only if it's run from the login shell.
I would not recommend sourcing .bashrc in the script. Instead, create a separate file named something like .script.init.sh and source that:
# script init
TMP_STRING='tmp string'
Your script:
# script
. ~/.script.init.sh
If this value must be available to any process spawned by the script, prefix it with export :
# script init
export TMP_STRING='tmp string'