The event was won by Swiss Marcel Hug who set a new world record of 2:49.55 minutes.
Defending champion Prawat, 40, crossed the line in 2:50.20 and Putharet 2:50.68.
Prawat, the country’s most decorated Paralympian, has now won seven gold, eight silver and one bronze medals since the 2000 Sydney Games.
The Sa Kaeo native has one more event - the 800m T54 - left at the Tokyo Games which could be his last Paralympic race.
It is the 26-year-old Putharet’s second bronze at Tokyo 2020 as he also finished third in the 5,000m T54 event at the weekend.
Under the National Sports Development Fund’s bonus scheme for Paralympians, a champion will get B7.2 million, a silver winner B4.8mn, and a bronze medallist B3mn.
Thailand have so far claimed one gold, two silver and four bronze medals at the Tokyo Paralympics.
They aim to claim four to six gold medals in Tokyo, having won six gold, six silver and six bronze medals at Rio 2016.
The Thais are now pinning their hopes to reach the target on their athletes in boccia and wheelchair racer Pongsakorn Paeyo who won the 400m T53 title with a world record time on Sunday.
Elsewhere, Matt Stutzman said he would not be giving up on his quest for an elusive Paralympic gold medal after being knocked out in the last 16 for the second Games in a row yesterday.
“I’m not quitting,” said the London 2012 silver medallist after losing 143-137 to Slovakia’s Marcel Pavlik in the men’s individual compound open archery category.
“I’m going to be back in Paris  and my ultimate goal is to represent the United States in LA , that will be my last Games,” added Stutzman, who also fell at the same stage at Rio 2016.
The 2015 world champion, who was born with no arms and holds the bow with his foot, has an @ArmlessArcher Twitter handle where his profile declares proudly “I do everything with my feet”.
But the usually ebullient Stutzman cut a bitterly disappointed figure after a second successive early exit at the Paralympics and one of his “worst scores in five years”.
The 38-year-old American is one of the world’s most-recognised Paralympians after being featured in the acclaimed 2020 documentary “Rising Phoenix” alongside eight other athletes.
Unusually for someone who has handled his growing fame with grace, confidence and a great sense of humour, he confessed that it was nerves that had got the better of him.
“Everything felt good in practice but I let the adrenaline get to me,” he said.
“My bow just doesn’t stop moving. I can’t calm it down, I can’t control it. I felt like crap. That’s the polite way to say it. That was one of my worst scores I’ve shot in probably five years.”
China’s He Zihao, who set a new Paralympics record score of 148 in the quarter-finals, won the gold with a 147-143 victory over Iran’s Ramezan Biabani.
Pavlik missed out on a medal when he was beaten 144-142 by An Xinliang of China in the bronze match.
Additional reporting by AFP