You have the basics right, but it doesn't quite work like you presuppose in your question.
First of all, you're right that the President can pledge to veto a budget if it doesn't meet his requirements. Usually, there's already been a lot of negotiation between the White House and the leaders from both parties in the House. It's only when that process reaches an impasse that the President would issue an ultimatum like this.
You're also right that if Congress passes a budget that the President vetoes, it would take 2/3 of both houses to override it. And the likelihood of being able to override a veto plays into the negotiations, too.
However, it's not just about the bottom line totals. There's so much that goes into the federal budget that it doesn't really make sense to say "This is the maximum dollar value", and there's definitely no correlation between the total value and whether or not the 2/3 majority can be reached.
For example, in 1995 and 1996, the government shut down when Clinton vetoed a spending bill after negotiations broke down:
Since a budget for the new fiscal year was not approved, on October 1 the entire federal government operated on a continuing resolution authorizing interim funding for departments until new budgets were approved. The continuing resolution was set to expire on November 13 at midnight, at which time non-essential government services were required to cease operations in order to prevent expending funds that had not yet been appropriated. Congress passed a continuing resolution for funding and a bill to limit debt, which Clinton vetoed as he denounced them as "backdoor efforts" to cut the budget in a partisan manner.
It was eventually resolved:
The first budget shutdown concluded with Congress enacting a temporary spending bill, but the underlying disagreement between Gingrich and Clinton was not resolved, leading to the second shutdown.
The second shutdown lasted 22 days as White House and Congressional negotiators worked out a balanced budget agreement that included modest spending cuts and tax increases.
You'll notice here that the resolution didn't include a flat adjustment - instead, it was a whole package of changes that everyone was able to agree to.
The government shut down again in 2013 when the two houses of Congress failed to agree on a bill, which is a slightly different situation with the same effect. Again, that was resolved by give-and-take negotiation (although it ended up being mostly one-sided), not adjusting it to meet some arbitrary dollar value.