Yes you can deal with the third parties Insurance company directly, They will be in the same position as your own company in that they won't repair the car, if the repairs are high in comparison with the cars market value. So you would still be faced with a write off. The benefit of dealing with the third parties Insurer is that they won't deduct an excess from the write off payment they make. You might be able to negotiate having the salvage returned to you, with the relevant amount deducted from the claim payout. If you can then get the car repaired yourself out of the claim payout, then that may be a way of keeping the car. Be aware that if you claim from the third parties Insurers, you cannot use the FOS to help with any complaint.
You may be able to do the same with your own Insurers. Ask them to return the salvage, deduct the relevant amount from the claim payout and then you can sort out the repairs yourself. The problem with that option, is that they would deduct the excess and you would have to wait to get this back from the third parties Insurers.
Suggest that you discuss this with your own Insurers first, as sometimes getting salvage back to repair yourself, is not always an option. It depends on the level of damage.