Effect of Bill
First of all, if this bill (Idaho House Bill 675) is passed in the form linked in the question, section 4, which prohibits transporting a child outside the state, ONLY applies when this is done for purposes prohibited in subsection (1), that is, traditionally defined female genital mutilation. It does not, in its linked form, prohibit transport to violate the new section (2) which prohibits procedures to "change or affirm the child's perception of the child's sex". Thus the statement in the question that:
subsection (4) of the bill purports to criminalize transporting a minor out of state (to neighboring Oregon, say) for the provision of such treatment.
is incorrect.
Notice that section 4 is not new, nor is it modified (except by changing th4e section number) by the current bill. My understanding is that the travel ban was, when it was passed, aimed at families who took female children outside the US, to African or Middle-eastern countries, for traditional rites that include FGM. This was done by a significant number of people with ethnic origins in such countries, and a number of US states passed laws banning such travel. This law, before the house bill, was one such law. The recent bill added "gender affirming care" into the law, by calling it another form of genital mutilation.
However, for purposes of discussion, let us assume that subsection (4) is changed to refer to subsections (1) or (2), thus making the question correct.
Right to Travel
The right to travel from one state to another is a basic right under the US constitution.
The LII page on "Interstate Travel" refers to it as: "a right venerable for its longevity, but still lacking a clear doctrinal basis."
See the Wikipedia article Freedom of movement under United States law
The right to travel was recognized early on, in the case of Corfield v. Coryell (6 Fed. Cas. 546, no. 3,230 C.C.E.D.Pa. 1823). This case was decided by Justice Bushrod Washington, a member of the US Supreme Court, but it was decided while he was acting in his role as circuit judge, not by the Supreme Court as a whole.
In that decision, Justice Washington included "The right of a citizen of one state to pass through, or to reside in any other state, for purposes of trade, agriculture, professional pursuits, or otherwise;" as one of the "privileges and immunities of citizens in the several states" which are guaranteed to all citizens. This passage has been influential, although not undisputed, in interpreting the "privileges and immunities clause" of the Fourteenth Amendment.
In Crandall v. Nevada, 73 U.S. 35 (1868) The Supreme Court held that the right to travel between states was fundamental, and that a state could not impede it by taxing the exit of people from one state to travel into another.
Subsequent cases have deemed the right to travel protected against state restrictions both under the privileges and immunities clause, and under the commerce clause. In United States v. Guest, 383 U.S. 745 (1966) at 383 U.S. 757 the Court wrote:
The constitutional right to travel from one State to another, and necessarily to use the highways and other instrumentalities of interstate commerce in doing so, occupies a position fundamental to the concept of our Federal Union. It is a right that has been firmly established and repeatedly recognized.
The Guest court goes on to cite Crandall v. Nevada, 6 Wall. 35, (the case in which a tax on those leaving the state was overturned) and the Passenger Cases, 7 How. 283, 48 U. S. 492 in which Chief Justice Taney wrote:
We are all citizens of the United States; and, as members of the same community, must have the right to pass and repass through every part of it without interruption, as freely as in our own States."
All this would mean that a law prohibiting travel would need strong justification, and is probably subject to strict scrutiny, a standard of review requiring a "compelling governmental interest" and a "narrowly drawn" law.
Medical Judgement
The bill declares that procedures to "change or affirm the child's perception of the child's sex" are "never necessary", any medical opinion to the contrary notwithstanding. Such an override of qualified medical opinion might be hard to justify in a criminal law.
Civil Rights
In BOSTOCK v. CLAYTON COUNTY, GEORGIA 590 U. S. ____ (2020) #17–1618 the Subreme court ruled that Title VII of the Civil Rights Act of 1964. prohibits employment discrimination against homosexual or transgender individuals, writing:
Because discrimination on the basis of homosexuality or transgender
status requires an employer to intentionally treat individual employees differently because of their sex, an employer who intentionally penalizes an employee for being homosexual or transgender also violates
Title VII.
This rule may well carry over to other areas of the law.
Equal Protection Clause
The Equal Protection Clause of the Fourteenth Amendment reads
nor [shall any state] deny to any person within its jurisdiction the equal protection of the laws.
Treating members groups differently because of group membership is often, but not always, an equal protection violation. The provisions oc the bill, if enacted, might constitute such a violation.
