Wouldn't remove the filter, yes it will work but could clog up again as large lumps could then reach your pump.
If your pump is making a noise then its likely just the filter clogged, if no noise then the pump is gone more than likely.
Yes remove wheel, remove rear part of the arch liner, probably about 10 screws and bolts. One bolt frees up the bottle to slip forward and the bottle can be easily accessed. Have a bucket ready as there is lots of water. There are 2 pumps and filters, one for headlight washers and one for windscreen, worth cleaning them both when you are there although likely the headlight one won't be too bad. Neither of mine looked severly clogged up but after cleaning have a lovely strong squirt now.
And I filled with BMW screenwash to allegedly avoid the issue reoccuring.