If I were him I would make a show of it and bring cameras and media.
I would calmly walk into the Senate building greeting the Senators and workers.
I would cross under the Rotunda and do a little tour and show the audience. I think you can see the old Sup Ct room down there... if I recall.
In that exact area... I would talk about history. Founders... checks and balances. Providing President to flex his power and the agenda upon which he was elected with the veto. I would explain this showdown was caused by Congress and their omnibus budgets and spending bills. I would explain I should have been about to veto numerous spending bills in the past to force Dems to the table to negotiate... but because they go omnibus... I have to use the power the Founders granted the executive on whatever bills I can.
He should explain all this is a dysfunction of Congress.... partially caused by gerrymandering. Hence many congress man are secure in their lopsidedly left districts. Congress should be really feeling this pain of this shutdown.
But they are being dysfunctional.
I would talk about the founders setting up checks and balances and thereby giving the President the veto. Which forces Democrats to work with me or override my veto..
What a historical opportunity to drive a Presidents points home.
Then I would slowly walk over to the house. I would have my Secret Service people making sure that the the Capitol Police and the Sargent at Arms people knew that they could not touch Trump or the Federal People surrounding him.
I would have few Republicans open the door for me.
I would then with my entourage march up on the Podium. I would allow the news people to place their mics and cameras and I go into my second teaching moment and give a great state of the Union but say the swamp still worries you. Obamacare still worries. Taxes are still to of a burden. Our borders too insecure... etc.
And commence.
sorry for grammar and typos. I have to do something else now.
This sets the table where he can give the address elsewhere and make a big fuss about obstructionist Democrats.