The budget is technically like any other bill, so the president is allowed to veto it. This tends to have severe political fallout, so it doesn't usually happe

The budget is technically like any other bill, so the president is allowed to veto it. This tends to have severe political fallout, so it doesn't usually happen, but it can. He also generally has a lot of unofficial influence with legislators in his party, so he can use that influence to push things one way or another. That's usually the main effect, and Trump would likely just use his influence on the senate here.