Some alternative "defenses" the White House might try if "constitutional immunity" doesn't fly:

"the Mothership has instructed us not to testify";

"the planets' alignment contra-indicates subpoena compliance";

"no one in America except the Democrat snowflakes cares if we are treason-weasels";

"it's a grand old flag!";

"Washington and Lincoln never got subpoenas";

"this falcon cannot hear the falconer";

"but---her emails!"

All the above are just as valid and relevant as the "we just pulled this out of our ass" defense of "constitutional immunity".