A bill that's passed both houses then needs Royal Assent from the Queen to pass into law & by constitutional convention the Queen always follows advice of ministers (the PM) on these matters.
It appears to be unequivocal (them's the rules) if controversial (a lot of people don't like it) that this is the case & there's nothing parliament can do about it, for the purpose of this question I'm assuming that is the case & the PM does it. If you have any comment to the contrary please make them in comments rather than an answer.
If the PM does advise the Queen not to give Royal Assent to the new Brexit delay bill, aka the European Union (Withdrawal) (No. 6) Bill, then the only remaining route to stop Brexit seems to be a vote of no confidence to force a new election.
However, if no clear alternative government has or can form a majority in the house (& at this point it seems implausible in the extreme Corbyn or any other could negotiate enough support from other groups to do so) then it's GE time.. that leaves Boris as PM & the PM sets the date for the GE.
Is there anything to stop him setting the date of the election after 31 October if that happens?