You do realise that the executive office has it’s own lawyer for the President, Elena Kagan is his Solicitor General and is tasked with representing him.
No, the Solicitor General's primary responsibility is to represent the United States before the Supreme Court. You may be confusing her with the White House Counsel's office. That person is the principle legal advisor to the president on policy questions - conflicts of interest, ethics questions, vetoes, things like that.
Again, the president gets sued dozens, if not hundreds of times each year. I was reading not too long ago where Obama was sued by a bunch of kids in LA because their parents were deported as illegal aliens. Neither the White House Counsel or the Solicitor General has the time or staff to deal with all the nuisance suits that get filed.
The US Solicitor General is the representative for the US Government in cases that come before the US Supreme Court, nothing more nothing less.
They will either argue the case that's brought by and for or against the US, or they will file amici curiae briefs on behalf of the US government in cases in which there are special interest or concerns of the US government.
The Solicitor General is in no way characterized as the President's own lawyer.