After a jubilant morning rally on the steps of the Capitol to debut their health care bill, House Democrats on Thursday returned to the hard work of resolving remaining internal disputes over its final shape — including a nagging question about illegal immigrants’ access to care.The hot-button debate animated town hall protests in August and prompted Rep. Joe Wilson (R-S.C.) to shout “You lie!— at President Barack Obama when he addressed it in a speech to Congress last month.It’s remained relatively quiet since. But now, with the fight over the profile of the public insurance option settled, it is quickly re-emerging along with a dispute over abortion language as the next major headache for Democratic top brass.House leaders are struggling with whether to match a tougher Senate approach, developed in response to the Wilson episode, that would bar undocumented workers from using their own money to buy insurance in a new health care exchange.The Senate’s hard line has the backing of the White House — and of politically vulnerable House Democrats who assume a final package will include the strictest version. Leading the charge for it in leadership is Rep. Chris Van Hollen (Md.), the official keeper of endangered Democrats as Assistant to the Speaker and Democratic Congressional Campaign Committee Chairman. He is trying to tighten the House language in a manager’s amendment set to roll out early next week.But members of the Congressional Hispanic Caucus are pushing back, and they have the backing of some top Democrats who argue the change amounts to bad public policy. Among them: Energy and Commerce Chairman Henry Waxman (D-Calif.), a key architect of the House bill.“Do we say illegal immigrants can't buy a car? ...Do we say illegal immigrants can't buy food that has a federally mandated label on it?" he said. Waxman said he was surprised at the White House position and hopes it changes, pointing to “pretty strong— House language that prevents illegal immigrants from accessing subsidies.Members of the CHC were playing their position close to the vest on Thursday in the apparent hopes of not drawing too much attention to an issue that has the potential to electrify right-wing opponents.CHC Chairwoman Nydia Velázquez (D-N.Y.) said her group is “supportive of the bill that’s been introduced.— She said advocates of the tougher approach “should know better— than to try to push it in the coming package of amendments. But of such a change, she said, “We’ll cross that bridge when we come to it.—On the immigration front, the debate over access to the exchange appears to be the leading concern. But there is also some discussion of whether to roll back a five-year prohibition on legal immigrants getting any federal benefits and tweak the verification provisions already included in the bill. Steven T. Dennis contributed to this report.