I have a bash script that calls/invokes a sub-script with command-line arguments. How can I force the sub-script to run under the same PID as the caller?

Alternatively, source the sub-script:

one.sh

#!/bin/sh

echo one.sh: pid is "$$"
. ./two.sh
echo done with "$0"

(the . command is exactly the same as source in bash, but . is more portable)

two.sh

echo two.sh: pid is "$$"

Sample run:

$ ./one.sh
one.sh: pid is 31290
two.sh: pid is 31290
done with ./one.sh

The script two.sh will be run in the same shell environment as one.sh, and the shell will not spawn a new process to run it. It behaves very much like calling a shell function, in more ways than one (for example, use return rather than exit to return control from two.sh to one.sh early; exit would exit the shell session completely).

If you don't need to run any other commands or do anything else after calling the sub-script, then you can use:

exec sub_script

The exec is a shell builtin which does:

$ help exec
exec: exec [-cl] [-a name] [command [arguments ...]] [redirection ...]
 Replace the shell with the given command.

 Execute COMMAND, replacing this shell with the specified program.
 ARGUMENTS become the arguments to COMMAND. If COMMAND is not specified,
 any redirections take effect in the current shell.

 Options:
 -a name pass NAME as the zeroth argument to COMMAND
 -c execute COMMAND with an empty environment
 -l place a dash in the zeroth argument to COMMAND

 If the command cannot be executed, a non-interactive shell exits, unless
 the shell option `execfail' is set.

 Exit Status:
 Returns success unless COMMAND is not found or a redirection error occurs.

So after running exec sub_script you have permanently left the parent script and cannot go back to it again.