You must separate the error on the right hand side of an assignement form the command used. If you want a "one line solution" for the declare command, you can use:
value=$(false); declare -r var="$value"
If you do not want to use a new var, you can use builtin ephemeral vars like either $REPLY (used by read) or $_ (the last argument of a previous command line).
Using $REPLY might be as simple as:
REPLY=$(false); declare -r var="$REPLY"
Using $_ may become complex or convoluted.
For me: Using set -e has proven to be more of a problem than a solution. Avoid it.