The OP should never have mentioned the other damage because its just confusing everyone. Lets go through this step by step.
The OP's car is damaged by a third party.
The third partys insurance company has a duty to put the OP back in the position he was in before the incident occured. This means they have a duty to repair his car, ensure he is not disadvantaged whilst the repair is carried out, so on and so forth. This means that the insurance company must pay for:
a) A hire car
b) Bodyshop
c) Transport of the car and from the bodyshop if appopriate
d) Any other expenses
With me so far? Thats the insurers obligation.
The insurer has decided to approach the OP and offer to pay cash compensation IN LEIU of meeting its obligation to put him back in the position he was in before. They are basically saying 'Look, we know we have to do Y, but how about if we just offer you X, and then we are both happy?'. They are doing this because if they can get the OP to accept the money, it will cost them less than meeting the obligations they have. And, provided both parties are happy with the offer, this is fine and legal and absolves the insurer of its responsibilities.
The OP is waiving his entitlement and receiving a cash offer instead. This is entirely the insurers choice to offer this method of settlement, just as its entirely the OP's choice to reject this and insist instead that they put him back in the position he was in before.
Get it yet?
Right.
Now. Lets go into the OP's mind.
He wants his car fixed. The line of least hassle is to say 'Please fix my car'. The insurance company will pay for the repair, and a hire car, causing the OP the least amount of hassle. The OP has now been offered cash instead. Cash on its own is no good to the OP because before the accident he had a non damaged car, not a pile of money. He takes on hassle as a result of accepting the cash offer (Which is being made to save the insurer money) and therefore it must be worth his while. He now has two seperate thought processes that people are unfairly linking.
Thought process 1: I need to make sure the offer they give me is as much in my favour as possible, if they are going to get out of their obligation to simply sort evreything out for me.
Thought process 2: This could come in handy because whilst its in the bodyshop, I could get some extra work done, using some of the compensation they are encouraging me to accept instead of excercising my right to be put back in the position I was before the accident.
This is different to Slime101's situation because:
Slime101 has been offered by the council to have his car put back in the position it was in before the accident. This is the obligation they have. They have chosen not to make a cash offer in lieu of settlement. Slime101, however, wishes to get them to give him cash instead so he can not bother to get the car fixed and can instead spend the money on something else.
The reason its different is because the two situations are opposite way rounds. It is not the for the innocent party to dictate whether the insurer makes a cash offer in settlement.
The above is, as always, simply my opinion. I am not a lawyer.