As a thought - what if you gave up The Great Experiment and implemented a Westminster system instead...
The President would become a ceremonial figurehead (directly elected by popular vote or a McGarvie model -see below), sign the bills and maybe still hold but rarely use veto power.
The President would become a ceremonial figurehead (directly elected by popular vote or a McGarvie model -see below), sign the bills and maybe still hold but rarely use veto power.
https://en.m.wikipedia.org/wiki/McGarvie_Model
Disclaimer: Australian so obviously biased