Can Alice still get child support, or will she be denied because she
can not definitively prove which man is the child's biological father?
The legal standard is a preponderance of the evidence (i.e. more likely than not) and there is plenty of evidence that can be offered in addition to DNA evidence, such as testimony under oath from people in a position to know who was having sex with whom at the relevant times.
Contrary to a common misconception, testimony under oath is still solid evidence that can support a verdict on appeal.
Alice had neither a long standing romantic relationship with Bob nor
anyone that was a witness to the sexual act, thus making the question
of who she slept with difficult to prove.
It isn't that hard to prove.
Q to Alice's physician: Based upon an ultrasound, when did Alice
conceive?
A: April 5-8, 2021.
Q to DNA expert: Based upon the DNA test, who could the father be?
A: Billy or Bob.
Q to Alice: Did you have sex with Billy between April 5-8, 2021?
A: No.
Q to Alice: Did you have sex with Bob between April 5-8, 2021?
A: Yes.
Q to Billy: Did you have sex with Alice between April 5-8, 2021?
A: No.
Q to Billy: Why not?
A: I was at the Shuffleboard World Cup in Tibet, I have time stamped
pictures.
Q to Bob: Did you have sex with Alice between April 5-8, 2021?
A: -- if Yes, judge says he believes Bob and Alice and the case is
over. -- if No, the judge decides who among Bob, Billy, and Alice
the judge believes based upon other evidence.
Ultimately, the judge has to rule between the two based upon non-genetic evidence and resolve credibility disputes just as in any other case that doesn't involve DNA evidence (which is the vast majority of cases).
Also, the edge cases are few are far between. Identical twins are rare to start with, and few women have sex with more than one identical twin in the several day period when she could have conceived or didn't know which twin she had sex with. It has happened at least once in history (post-DNA testing), but you can probably count the number of times that it has ever happened on one hand.
For example, presumptions from cohabitation, marriage, and claims of paternity often resolve paternity disputes without DNA evidence.
Further, to the extent that there is good faith uncertainty (perhaps everyone agrees that the mother has sex with both twins on the only possible day of conception and nobody really knows), the downsides to a mistake in the larger cosmic sense of the overall paternity law system are minimal, as identical twins very rarely become deeply alienated from each other and instead tend to be close and intensely cooperative once they discover each other, and tend to be similar to each other in almost every respect depriving the child of little if the court gets it wrong. Realistically, identical twins are particularly likely to settle out of court so the judge doesn't have to decide.
In one of the only two actual cases I could locate that went to trial (in Brazil), both twins were ordered to pay child support because the evidence showed that they actively conspired with each other to confound the mother and the court regarding who the father was, and conspiracies can support joint and several liability.
The other case reported in a news story had convincing circumstantial evidence supporting one identical twin over the other that probably establish a presumption of paternity for one twin and not the other.
One of the twins, who cannot be named for legal reasons, went to court
last summer in the hope of forcing the mother to grant him access to
the child. Although his name is not on the birth certificate, he
claims he is the only father the boy has known, cared for him every
other weekend, provided financial support and was even known to him as
'papa'.
But then the man's relationship with his girlfriend broke down and the
visits halted. When he began legal proceedings to prove his paternity,
the mother made her claim that she had been sleeping with his twin at
around the same time.
The twins have said they knew they were both having sex with the
woman, but argue that only one had sex during the period of
conception. Both refused to undergo a DNA test: the complainant
refused to pay the £335 charge while his brother, who has since
married and fathered children, does not consider himself involved in
the dispute.
Now, however, Judge Jolin has asked the complainant to take a DNA test
by 1 December to ensure he can claim even possible paternity, while
his brother may also be tested.
(The second case is in Quebec and the cost of the test in pounds is apparently a currency conversion value.)
(It is possible in principle to distinguish even identical twins from each other with high coverage whole genome tests that would reveal a few random mutations in each twin out of billions of possible mutations, but it is currently prohibitively expensive to do so.)
Can she even get a paternity test given that it would not be
definitive proof which man was the father?
Yes. This rules out all 4 billion men in the world minus two of them. It has great probative value, narrowing the list of possible fathers down to two.