Voting rights today
The House and Senate may set the qualifications for voting for themselves, except that the amendments to the Constitution require that women, people 18 and over, and blacks, be allowed to vote equally with older white men, that is, they can't be singled out.
So a law passed in 1865 would have had to be overturned by a House and Senate vote, and by the concurrence of the President , rather than by the action of individual states. It could have been done, and it could have been undone by a later House and Senate and President.
Article One, Section Four
"The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing Senators."
That last is because the Senators were originally chosen by the state legislatures.
Article One, Section Seven
"every Bill which shall have passed the House of Representatives and the Senate, shall, before it becomes Law, be presented to the President"
That would have been hard to do. It needs a majority of both Houses and the President all at the same time. Not impossible, but certainly difficult to do legally.
The interesting thing is that in 1864 the Radical Republicans could have rammed one through letting everybody vote, blacks, women, and hell, kids. They still could.
Maybe in 1992 when Clinton had majorities in both houses?
What if the Republicans decide that the proper thing to do is to disenfranchise everybody that doesn't make at least 100,000 a year?
No, that wouldn't work. High earning people are socially more liberal than low earning people. The Moral Majority is only half the Republican party but they absolutely control it.
Abolish voting rights for people that aren't married? A religious test for voting isn't constitutional, but if it doesn't specify which religion....