The U.S. Senate voted in January 2019 to block the Born-Alive Abortion Survivors Protection Act, which in the words of the bill itself would prohibit:

a health care practitioner from failing to exercise the proper degree of care in the case of a child who survives an abortion or attempted abortion.

Fox News reports that the bill was blocked by Democrats in the Senate, including all of the Democrats currently running for president in 2020 (I could not find any coverage of this vote on CNN). The Fox News article mostly discussed the Republican support for the bill, but it does have some information about why Democrats opposed it:

Opponents, noting the rarity of such births and citing laws already making it a crime to kill newborn babies, said the bill was unnecessary. They said it was part of a push by abortion opponents to curb access to the procedure and intimidate doctors who perform it, and said late-term abortions generally occur when the infant is considered incapable of surviving after birth.

“This bill is just another line of attack in the ongoing war on women’s health,” New Hampshire Democratic Sen. Jeanne Shaheen said on the Senate floor.

...Sen. Patty Murray, D-Wash., objected to Sasse's bill, saying the legislation was unnecessary and amounted to a political stunt.

However, the article also recalls recent actions by Democrats such as Virginia Governor Ralph Northam and Virginia Delegate Kathy Tran, the latter of whom sponsored a state bill to allow third-trimester abortions and the former of whom endorsed it by saying

"When we talk about third-trimester abortions, these are done with the consent of, obviously, the mother, with the consent of the physicians, more than one physician, by the way," Northam said. "And, it's done in cases where there may be severe deformities, there may be a fetus that's non-viable."

Northam continued: "So, in this particular example, if a mother is in labor, I can tell you exactly what would happen. The infant would be delivered. The infant would be kept comfortable. The infant would be resuscitated if that's what the mother and the family desired, and then a discussion would ensue between the physicians and the mother. So, I think this was really blown out of proportion."

The Born-Alive Abortion Survivors Protection Act seems to have been introduced at least in part in response to the suggestion that a live infant who survived a late-term abortion might not be resuscitated.

If Democrats at the state level are introducing legislation to allow late-term abortions then why are Democrats at the federal level claiming that late-term abortions are so rare that this regulation of such abortions is unnecessary? Is this a just a disconnect between Democrats at the state and federal levels?

I'm also not sure what Senator Shaheen's comment about "women’s health" has to do with the bill, since the law deals with infants who have already been born. Additionally, the law would prohibit prosecution of the mother who sought the abortion and allow the mother to obtain "appropriate relief" in a civil action against the physician who violated the law (e.g. "statutory damages equal to 3 times the cost of the abortion or attempted abortion").

Have any senators who voted against the bill provided a detailed argument for their opposition to the bill beyond the sound bites quoted above?

Technically infanticide is already illegal

Roe v Wade legalized abortion. Technically, you cannot abort a fetus once it has left the mother and Federal law prohibits it. Kermit Gosnell, a late term abortionist, was charged with (but not convicted of) one count of infanticide, an assertion by members of his staff (testifying against him) that some fetuses from late term abortions were still moving, and Gosnell murdered them.

Sen Tim Kaine stated

Sen. Tim Kaine (D-Va.), who is Catholic, released a statement after the vote saying he opposed the bill because GOP statements about it are "misleading."

"Congress reaffirmed that fact with its passage of the bipartisan Born-Alive Infants Protection Act in 2002. I support that law, which is still in effect. There is no need for additional federal legislation on this topic," Kaine said.

Democrats fear it will lead to undue abortion provider scrutiny

Sen Chuck Schumer said this

Senate minority leader Chuck Schumer (D., N.Y.) said on the Senate floor that the born-alive bill “is carefully crafted to target, intimidate, and shut down reproductive health care providers.” He also claimed the bill “would impose requirements on what type of care doctors must provide in certain circumstances, even if that care is ineffective, contradictory to medical evidence, and against the family’s wishes.”

Democrats may feel it might be part of a broader push

From Vox

The bill may also be part of a larger strategy by Republicans of focusing on very late abortions in order to drum up support among social conservatives. Trump referenced the issue in his State of the Union speech, saying, “we had the case of the governor of Virginia where he basically stated he would execute a baby after birth.”

As you described, there appears to be a disconnect in the late-term abortion views between the state-level Democrats in Virginia and federal-level Democrats in Washington. Interestingly, however, in reading your question, that's not the primary disconnect in my view.

The bigger and more meaningful division may be between the Republicans backing the bill (who are talking about newborn babies) and the Democrats (who are still talking about abortion). The bill does not appear to be about abortion, but about human beings outside the womb.

So, in consideration of that incongruity, and the fact that infanticide is already illegal, this bill may be pure political strategy by pro-life Republicans, who see an opportunity after the highly-publicized, "pro-infanticide" comments of Virginia's governor Ralph Northam and Delegate Kathy Tran, to portray Democrats as so extreme that they even support killing newborn babies. In short, this action may be no more than an effort to gain a political edge in the 2020 general elections.

Because this bill does not directly involve abortion, yet it is a tangential issue and the word "abortion" is in the title, it's politically difficult for Democrats, who receive major support from abortion advocates, to support it. In fact, for political purposes alone, it's very difficult for Democrats to support any legislation that (even remotely) suggests restrictions on abortion. If they don't oppose restrictions they risk losing campaign funding and sparking a primary challenge.

In terms of the Democratic presidential candidates, supporting this bill would be tantamount to political suicide. Any candidate who supports this bill would be moving themselves to the Right on the political spectrum, making themselves vulnerable to attack from the Left, where control of the nomination is believed to be. So any candidate perceived as anything less than extreme Left on this issue enables primary opponents to characterize them as weak on abortion and in alliance with Republicans. Either one of those attacks can be enough to sink a primary campaign. Therefore, all candidates must oppose the bill in order to remain... viable.

Because of the widespread support for abortion rights in the Democratic party, the presidential candidates face no risk at this stage in opposing this legislation. But in the general election, where the electorate is more centrist than primary voters in both major parties, this vote may come back to haunt the nominee. That seems to be the overall Republican play here.