The trainer has all but ruled out running the Juddmonte Oaks winner in the Darley Irish Oaks at the Curragh on July 13 as she needs a decent break between her races.
"I wouldn't rule it out, but I think it's looking increasingly unlikely she'll go
to the Curragh," said Beckett. "She's not very big, she's not very robust and she's only had three starts.
"There's the ground as well. Although it was quick at Lingfield (Oaks Trial), she probably is better with a little dig in the ground and my concern would be that at this stage of her life she'd run her race on the trip over. For all those reasons, at the moment we are edging away from it."
However, the Whitsbury trainer admits he would dearly love to win the St Leger at Doncaster on September 13. "I wouldn't rule that out at all," Beckett went on.
"Quite apart from the time scale – who knows what the ground will be like in mid September – she's obviously going to get a mile and six, or at least you'd have to be more than hopeful.
"The track would suit her, it's often a race that is not beyond the realms of possibility for the likes of her and I'd love to win it apart from anything else."
Beckett also has the Darley Yorkshire Oaks at York on August 21 at the back of his mind.
"The obvious race if we didn't take her to Ireland would be the Yorkshire Oaks," he said.
"Although that gives me some time between races I had been a little concerned that Lingfield to Epsom wasn't time enough for her, though it was four weeks.
"We've got plenty of time to work it out."
Bouguereau's season will now be geared towards the St Leger, as well, with connections ruing a chance that got away in Saturday's Vodafone Derby.
Peter Chapple-Hyam's colt was a 100-1 shot at Epsom and finished ninth, but his owner Andrew Black feels he would have been closer to New Approach if things had panned out differently.
"It was exciting, but like some of the other horses, I think he could have run quite a big race if the jockeys had been a bit more opportunistic," said Black.
"I think we had a horse who definitely stayed. We instructed Alan (Munro) to sit tight off the expected breakneck pace and stay on to see where we finished, but the pace just wasn't on.
"They were plodding on for the first mile and it went to the speed horses so I was a bit disappointed and felt we missed an opportunity, along with a few others.
"New Approach and Tartan Bearer were two who looked like they didn't have the stamina."
Looking further ahead, Black continued: "Bouguereau is essentially a Leger horse. I'm not quite sure what we'll do, but he's better than people think and at least proved he wasn't a mug horse at the weekend."
"I think (Chapple-Hyam) will look at the Great Voltigeur as his short-term target and run maybe in France in between."
TODAY'S BESTDUKE'S DOUBLEDelayed Approach (8.00 Southwell), nap and Timetable (4.00 Redcar).
DUKE'S TREBLEDelayed Approach, Timetable and Countrywide City (2.30 Redcar).
JULIUS'S DOUBLEHeywood (7.45 Chester), nap and Storyland (4.15 Salisbury).
FORMNAPCarniolan (3.15 Chester)
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