"The era of good feelings" during James Monroe's presidency is probably the best example I can think of, but that goes all the way back to the 1820s. He was virtually unopposed for a second term and Congress pretty much agreed on all legislation during that time period. However, it wasn't really a two party system being "bipartisan", but more of a one-party system since the Federalists collapsed and basically the only elected officials at the time were Democratic Republicans. It became a two-party system again when they had infighting amongst themselves, leading to a breakaway group called the "National Republicans". They eventually became the Whigs, and the remaining "Democratic Republicans" eventually became Democrats.
Yeah, that’s about right. Plus then you had the factions with the Jacksonian Democrats and Whigs on the issue of slavery, which necessitated the collapse of one of those parties as a result (the Dems could’ve also, only because the Whigs were weaker by the 1850s).
I forgot if it was Madison or Monroe (I think Monroe) who was moving in the direct of the Washingtonian ideal of there being no parties (especially after the collapse of the Federalists). Only because he was reticent about assimilating the old Federalists into the Jeffersonian Republicans that it failed and would later cause the 1820s splintering between Jackson and JQ Adams.