The home side went to stumps in a strong position on day two at the WACA at 4-219 in reply to Tasmania's 277.
On a drama-filled day, Hilton Cartwright hit 52 for the hosts before having to abruptly retire not out and leave the WACA to join his wife for the birth of their second child.
WA officials were still unsure on Monday night if Cartwright would return on Tuesday, and if Tasmania and the match referee would allow him to resume his innings.
After Cartwright's emergency exit Josh Inglis (72no) and Cooper Connolly (42no) took over, continuing their impressive starts to the summer.
But there was no such joy for Bancroft.
After recording a pair of ducks in the first round against Queensland, the opener was caught behind for eight soon after lunch.
Making matters more frustrating for the 31-year-old was the fact it came on a rough day for Australia's other Test aspirants, with Sam Konstas, Marcus Harris and Matt Renshaw also failing.
Konstas was lbw for two for NSW against Victoria at the MCG, before Harris fell for 16 in the second innings after being out for 26 in the first.
And Matt Renshaw was also lbw for two for Queensland against South Australia, as his hopes of returning to the Test team in the vacant spot at the top of the order continued to dim.
NSW opener Sam Konstas fell cheaply in the Sheffield Shield match against Victoria. (Morgan Hancock/AAP PHOTOS)
Those outs opened the door for Bancroft to ram home his case, after last week being selected in the Australia A squad to face India A alongside Harris and Konstas.
But after edging two consecutive boundaries through the cordon, the right-hander was hit on the helmet ducking into a Riley Meredith short ball.
He then fell shortly after, edging a ball outside off stump from Kieran Elliott through to wicketkeeper Jake Doran.
Bancroft has been the leading run-scorer in the Shield over the past two summers, and that alone should strengthen his case to open against India in Perth on November 22.Â
But a lack of runs now threatens to let him down when it matters most, after he was caught behind without scoring in his opening two Shield innings earlier this month.
Sam Whiteman (13) and Aaron Hardie (12) also fell to impressive 29-year-old late-bloomer Elliott (3-31), while Mitch Marsh scored just nine.
Earlier, spinner Corey Rocchiccioli was the main destroyer for West Australians, taking 4-70.
The offspinner had Tasmania's leading run-scorer Mitchell Owen lbw for 69, after the right-hander had previously slog-swept him for two sixes over mid wicket.
Doran also hit 66 for Tasmania, before he became one of paceman Brody Couch's three victims for the innings when he edged behind early on day two.
Marsh did not bowl any of the 87.3 overs for WA but is confident he will be able to fill the void left by Cameron Green's back injury with the ball this summer.
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