Agree with what everyone else has offered.... Puzzling indeed, and frustrating.
I presume that the printer worked in wireless mode with the previous modem router - whatever it was. (My understanding here is that there are two new bits of kit in the equation - EE Fibre Broadband "BrightBox2" router with dual output at 5.0Ghz and 2.4Ghz and a Netgear N300 Extender.)
Is it practical for the present to see if the printer will work wirelessly with the new modem / router - forgetting about the Extender for the duration of the experiment? (shift printer if possible to close proximity to the new router)
If this works, explore possibilities of using printer communicating with the router directly ignoring the Extender box option for the printer.
One way might be to use the old Belkin router as a wireless access point for the printer only, with the router in WAP mode placed in a favourable position for the printer and fed with cat 5 or 6 cable 'tween router and the rejigged 'WAP'.
This is a cumbersome option but it's one which I have used on a couple of situations to extend the range of a wireless network to a specific remote wireless network device. Of course it may not be practical in your situation. It also means tying up another wireless network channel and a whole lot of other potential complications, but it can usually be made to work.
The frustrating bit which I understand and share is that this should all work as anticipated and as implied in the nice advertising spiel.
I hope that you can work through to a satisfactory solution.
FOR