How accessible is it? Can it be approached from both sides?
If it's something that can be moved and worked on at a bench or similar?
Not really got an idea but those questions might help the next person along
Holesaw with extension rod, you will have to chisel the middle out a few times
Purchased too much already!That's what I didnt want to do reallyPurchased too much already!
Are you sure it's all wood?
Have you put a single hole through the entire depth to confirm?
Isn't a piece of wood of that size likely to be structural? Sticking a 4" hole through it might not be a good idea.
Yep.
It's a couple of pieces at least
If it's not one single piece of paper timber then using a hole saw shouldn't be too difficult?

If it's 200mm or more thick I would imagine the first question would be "is it it OK to drill a hole that big through it?" For a timbers of that size it may well be weight bearing. Where is this 200mm thick piece of timber?
For joists, I thought the rule of thumb was not to notch more than a 1/8 of the width, or 1/2 if you drill on the centre of the width. (id check that)
Are you just refitting a kitchen or is this part of a build that will be inspected by a building regs officer?It's above the external door from the kitchen on the side of the house. Essentially the kitchen needs an extractor fan put in to comply with building regulations.
Are you just refitting a kitchen or is this part of a build that will be inspected by a building regs officer?
Find a better place for it just through masonry and use ducting to the extractor. Personally I wouldn't want to be exiting a building above a doorway anyway due to lintel and load stresses there anyway.