What about a very limited suspension veto to compensate for the fact that the President is elected every four years and the Congress is renewed every two. Say if the President vetos a Bill it works like our current veto but the next Congress elected after the midterms could then pass an identical bill and override a veto with a simple majority. The logic being that the people have spoken on the issue by electing a new Congress that intends to pass said law. The people shouldn't have to wait until the next Presidential election to change Presidents in order to see the bill become law. So it would be rather limited in scope. It wouldn't carry over from one Presidential term to the next. So a new President isn't limited by his predecessors vetoes. New bills vetoed for the first time in the second half of a President's term would then take at least three Congresses to pass assuming the President is reelected and vetoes again or a new President is elected who likewise vetoes an identical bill during the next Congress.
The implications are complicated and interesting. I suppose it would change the 'lame duck' issue somehwat. If an unfavorable Congress comes out of the midterms they'd be overriding a Presidential veto rather easily (assuming the same majority was also in power the first two years and was earlier thwarted by a veto). But on the other hand a President in the last two years of his second term could thwart the will of the Congress with a veto on new legislation that it could possibly take another two election cycles after he's out of office to allow for the Congress to pass by simple majority depending on how subsequent elections play out. So it would certainly tie the productivity of a current Congress to the actions of the previous Congress in a way that might slow them down and negate the obvious advantage of a simple majority to override a Presidential veto.
I think it would really only strengthen the hand of a Congress when a President of the opposite party is narrowly elected because if they can simply maintain a majority after the midterms they could pretty much pass whatever they want. So I suppose in order to avoid becoming a true lame duck in the second two years of his term a President would need to win big and carry with him a favorable Congressional majority. At least that way if his party looses during the midterms he can still wield some power with a veto against whatever the new Congress tries to do. So it would, over the course of the full Presidential term, negate the effects of close Presidential elections where the winner squeaks by and the Congress is narrowly divided. If he has to 'use up' his veto in the first two years and his party can't take control of the Congress in the midterms (unlikely for an already weak President) then he's basically done on the domestic policy front for the rest of his term. The system has then course corrected, in two years rather than four, against a President who was only barely elected, and without the support of the peoples Congressional representatives in the first place. Domestic policy is wholly given over to the legislature to reflect the will of the electorate while the President carries on with foreign policy and more limited executive duties. The system then resets with the next Presidential election.