Historical data about the monastery and its construction are unknown. The 19th century map indicates no existence of a chedi or prang. The temple is not mentioned on Phraya Boran Rachathanin's map drafted in 1926.
Making an assessment off all the monastic structures, in the zone demarcated by Chikun Rd, Pa Thon Rd, Pridi Banomyong Rd and U-thong Rd is rather difficult, as the position and name of the structures varies on different maps. On a 19th century map there are 15 structures counted, while on the 20th century PBR map there are 13 mentioned. There is inconsistence in the names and the positions. Even a map drafted in the 90's by the Fine Arts Department, what I presume, based on excavations in the zone, shed no light on this matter. Positions of monastic structures can be asserted, but their ancient names will remain questioned forever.
The location of this temple on the 19th century map, coincides with the temple presently known as Wat Ho Rakhang.
Wat Trae is mentioned in the Royal Chronicles of Ayutthaya around 1663. There was a rivalry between the royal page Chai Khan and Phra Phetracha. Chai Khan boosted against King Narai (r. 1656-1688) he was superior in a certain game, hinting especially at Phra Phetracha. King Narai aware of the rivalry designated, a day for a contest in the elephant-horse chase (1). The game started near Wat Trae and the run went parallel along the Makham Riang Canal until Wat Nang. The first round was won by Phra Phetracha on horse back. Chai Khan, realizing he was losing face, skipped the second round and went home. 
(1) The game was a re-enacting of an ancient method of catching wild elephants. A horse acts as bait to anger the elephants, then leads them into a log trap. Upon level ground an elephant can overtake a horse; upon ascent the horse has the advantage.
 The Royal Chronicles of Ayutthaya - Richard D. Cushman (2006) - page 168-9 / Source: Phra Cakkraphatdiphong - Rivalry of Phra Phet Racha and Chai Khan.
"The next morning His Majesty held court and all of the marshals attended together. Master Chai Khan, a royal page and the son of a holy nurse, prostrated himself and said to His Holy Grace, “In the display of chase elephant and bait horse, outside the sole exception of the Supreme Holy Lord Omnipotent, there is no- one I am afraid of.” The Supreme Holy Lord Omnipotent was aware that Master Chai Khan was intentionally and maliciously comparing himself to Phra Phet Racha and that Phra Phet Racha was equally knowledgeable, and so He answered Master Chai Khan by saying, “You would each take a turn riding the chase elephant and the bait horse, wouldn’t you?” Master Chai Khan said, “I’ll ride the chase elephant first.” Phra Phet Racha was agreeable. When the designated day arrived, Master Chai Khan rode the premier elephant Phaya Sower of the Three Realms, standing six sòk and six niu high, and Phra Phet Racha rode the horse Mountain of Time, standing three sòk and two niu high. The arena was laid out in the vicinity in front of the Monastery of the Trumpets with the horse and elephant one sen apart. Phra Phet Racha reined his horse into a baiting display. Master Chai Khan drove his elephant and chased him on up close to the Bridge of Bricks at the Monastery of the Hides, and the elephant reached for him. Phra Phet Racha, seeing it almost upon his person, drove his horse into Little Spire Alley and the elephant was left behind. When it was the turn of Phra Phet Racha to ride the elephant, Master Chai Khan fled off to his home. Phra Phet Racha came in for an audience, prostrated himself, spoke to the Holy Lord Omnipotent and related the substance of that entire matter so the King would be informed of all the details. The Supreme Holy Lord Omnipotent said, “Weren’t you aware that that little Chai Khan is a soldier [only] in talk?”
Text & map by Tricky Vandenberg - November 2010 Reviewed May 2012