The fact that this has come up would be enough to put me off going ahead with the extension.
Depends on how badly you need the extra room, doesn't it?
The initial commitment is only to have an archaeologist attend the foundation dig. You'll have to pay for the archaeologist, so that adds to the cost of the project, but you can factor that in to your overall cost-benefit analysis. The cost shouldn't be huge, because the excavation for the foundations shouldn't take very long.
All going well, nothing is turned up except a few rusting bottle tops and the bones of a much-loved family pet buried by previous owners of the house.
At worst — and this is a very small chance — something is turned up which causes the archaeologist to determine that a full archaeological excavation is required before the development can proceed. It's only at this point that you need to take a decision as to whether to proceed with the development, given the addtional cost (and delay) occasioned by the dig, or just abandon the project, fill in the hole dug for the foundations, and write off the money already spent.
So, the initial decision you need to make is whether to risk spending the cost of digging the hole and paying for the archaeologist, given that there is a small (but non-zero) chance that what is found in the hole might mean the project becomes uneconomic and so does not proceed.
Be aware that there's always a risk that, when you start a construction project, something turns up that means the project costs more, or takes longer, than initially projected — you hit bedrock at a place where you weren't expecting to*, or you discover a limestone cave where your foundations were going to be**, or you find human remains***, or whatever. These are small, but non-zero, risks.
[* That one happened to me — small project, but the cost went up by about 20% as a result.]
[** Happened to a friend — cost him a fortune to have the cavity filled with concrete before construction could proceed. He didn't have a choice because the cavity was imperilling the existing structure.]
[*** Not so far, DG.]