Constitutional Convention Fanfic:
President's powers are changed to be more like the French Presidency.
House is changed to proportional representation on the state level using the Wyoming Rule (the smallest jurisdiction gets one rep and the ratio should not exceed that ratio by 50% or greater for other jurisdictions), or capped at 750,000 to 1, whichever is lower.
Congress has fixed 4-year terms from the formation of a new Government. Vote of No Confidence can bring down a Speaker and produce snap elections, after which the 4-year clock resets. Speaker must command the confidence of a majority of Congress (allowing for minority rule, but only when a majority of Congress consents to that arrangement. E.g. supply-and-confidence for minority rule rather than a formal coalition for majority rule).
With PR on the state level you'd have to ask the question of dividing up "marginal seats." Say Hawaii with its 2 reps, where do you draw the line that Republicans get 1 seat or Democrats get 2? What if the result is 65/35 D/R? Does 35% of the population really get 0 federal reps while 65% gets 100% of the representation?
My answer would be the 67% rule. If a party captures 67% of the share of a marginal seat, then they capture that seat (so in a 10-seat state, the minimum threshold for a party to get 1 rep would be 6.7% of the vote).
No single-member parties in Congress. If you are your only representative, you have to agree to caucus with at least one other congressperson or your seat will be given to the next eligible party on the list.
The House gains most of the Executive's power over domestic policy while the executive keeps power in foreign/military policy.
Supreme Court would stay mostly the same: Each President gets to pick one Justice (and the most senior justice must retire to do so upon the newly elected President's inauguration). Retiring justices can pick their own replacement. Impeached/dead justices have their replacement picked by a 3/4ths majority of the remaining court.
Amendments (different from amendments we currently have):
Elections amendment. When it regards elections, the rule for government law changes from "free speech" to "fair speech." Gov is allowed to regulate elections unless it can be proven that the law has a disproportionate impact on an established political party (that is, disproportionate to the party's influence, so the GOP couldn't sue over a regulation that hurts them, unless it hurts them to the point where their natural share of the vote would be suppressed). All elections are publicly funded based upon a nonpartisan electoral commission determining what is needed for a certain position in a certain jurisdiction (e.g. the congressional elections in Georgia will cost $20 million for the whole state and no more, each party getting a share of that based upon how they poll, while it would only cost $5 million in Montana). Outside money is strictly prohibited, though groups can advocate for partisan organizations under their own power, they are just not allowed to raise money for the purposes of doing so.
No right to bear arms, let each state figure out what works best on guns (as that's an actual issue where regional differences matter).
Free speech is maintained, but any speech designed to promote material harm to a group of people can be outlawed.
Senate is gutted of most of their responsibilities: they exist as a check on the President and can veto Presidential appointments, and can also veto laws, both based on a 2/3rds majority vote. Are responsible for impeachment. Due to their lower profile, they revert to being appointed by the States (under agreement between state legislature and state executive) for 8-year terms.
Most every right in the UN Declaration of Human Rights is so declared as a Right in the United States, and congress or state and local governments are required to pass legislation enabling the attainment of those rights.