The other Insurer told me I could be liable for costs if I don't claim through them, is this True?

In some cases, yes. If you don't let us, or the companies dealing with your claim know about the third party's offer, then you could be liable for some costs. The third party, by offering their services, expect the costs they agree to be fair and reasonable, and won't pay for any unreasonable costs, for example, paying for a hire car which is bigger than your current vehicle.
If you decide to claim with us, you won't be charged any additional costs (other than your policy excess) for making a claim for damages through your own comprehensive insurance policy with us. You're allowed to claim directly through the third party insurer and their solicitors if they offer their services to you at the appropriate time and you wish to accept their offer. Please contact us to discuss this first.