If Johnson had really wanted to escalate the Vietnam War, he could have made greater spending cuts domestically. He tried to fund the Vietnam War, and his ambition “Great-Society” welfare reforms simultaneously, meaning neither got the budget required to make the policies a success. 
Had he have made the Vietnam War his sole priority, he would have had significantly more financial resources available.
Also, Congress permitted exceptions in cases when it restricted funding. For example, the McGovern-Hatfield amendment of 1970 only prohibited funds for certain military activities, including escalating troops.
The President had plenty of political space to pursue other military avenues aside from escalating troop numbers.