Yes. The WH will do that just to protect the institution of the WH, has nothing to do with Trump.
If this were the actions of a Senate committee/subcommittee wanting testimony from administration members then yes, Trump asserting executive privilege is proper. But this is a trial venue and not a legislative one with Senators as Jurors and not legislators. It's possible that executive privilege doesn't apply here and any attempt to assert it could earn the President a contempt of court citation from the presiding Chief Justice.
totally uncharted constitutionally. Doubt that could happen.