WREXHAM lost 4-3 on penalties as they drew 2-2 after 90 minutes against Altrincham who reached the FA Trophy fifth round.
Altrincham made a lively start at Moss Lane and were on top when they took an early lead but Bickerstaff, who found the back of the net in the previous round win against Scunthorpe, bagged a quickfire brace before the break.
It was an end to end fourth round encounter and Wrexham conceded a late leveller before Altrincham won the shoot-out where Sam Dalby, James Jones and Harry Lennon scored their spot-kicks but Bickerstaff and Reece Hall-Johnson missed.
Manager Phil Parkinson said he would name a different side to the one that started Tuesday's 2-1 win against Bromley which maintained the National League title challenge and the Reds' boss was true to his word.
He still named a strong team with youngsters making up the substitutes bench as all regulars were given an evening off.
Altrincham put Wrexham under early pressure and Jordan Hulme fired into the side-netting for the home side.
There was no let-up from Altrincham who took a 10th minute lead.
Wrexham failed to deal with a corner, Chris Conn-Clarke struck the crossbar from just outside the area and Lewis Baines converted the rebound.
It was almost 2-0 when Baines looked like he had scored again but Marcus Dinanga got a touch on the line and the goal was ruled out for offside.
Altrincham were well on top and Rob Lainton pulled off a brilliant save to turn Hulme's header onto the bar and Reece Hall-Johnson made a vital block to deny Toby Mullarkey.
Wrexham had their first sight of goal when David Jones played in Sam Dalby and the striker brought a comfortable save out of keeper Ollie Byrne.
Bickerstaff also had an effort blocked by Byrne but Wrexham equalised in the 24th minute.
A cross was tipped away by Byrne but James Jones' shot was turned home by Bickerstaff.
It got even better as Wrexham went 2-1 up on 27 minutes.
Dalby found Bickerstaff who ran through at goal and calmly fired past Byrne.
Andy Cannon saw his drive easily saved by the goalkeeper while Hulme fired into the side-netting for Altrincham as the game switched from end to end.
Lainton denied danger-man Hulme with the last chance of the first half, while Wrexham twice found the side-netting through Cannon and James Jones when the action resumed.
Conn-Clarke only just cleared the bar for Altrincham and Egli Kaja shot straight at Lainton.
Bickerstaff - on a hat-trick - had an effort parried by Byrne as Wrexham responded and James Jones tried his luck from outside the area but was thwarted by a brilliant Byrne save with his effort heading for the top corner.
Liam McAlinden was only just too high with a powerful drive from the edge of the box while ex-Wrexham captain Connor Jennings shot straight at Lainton for the home side.
However, Altrincham equalised in the 89th minute.
Wrexham failed to clear their lines and substitute Zak Goodson fired home to take the game the game to a shoot-out where Altrincham won to reach the fifth round.
Altrincham (4-3-3): Byrne; Mullarkey, James Jones, Baines, Eddy Jones; Lundstram, Conn-Clarke, Marriott; Hulme (Pringle 67), Dinanga (Jennings 75), Kaja (Goodson 58). Subs not used: Osborne, Gould, Jordan.
Wrexham (5-3-2): Lainton; Hall-Johnson, Cleworth, Butler (James 85), Lennon, McAlinden; James Jones, David Jones (Austin 78), Cannon; Bickerstaff, Dalby. Subs: Calderbank-Park, Cushion, Dan Davies, Dan Jones, Milner.
Attendance: 2,526 (985 from Wrexham).
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