I think most are misunderstanding. The way that i read this is the at fault driver is saying we will pay for the deductible and rental as long as you have your insurance company process the claim. Meaning they will make sure repairs are equitable, done right and then process the claim through the at fault driver. Remember the OP has to requirement to process the claim through their insurance, they can do it all on their own, take the check, cash it and never repair or eventually come back and sue the at fault driver.

Sounds to me the at fault company it taking the lead to push this through insurance and overall help limit any extra paperwork or hassle that may arise from someone not going through insurance to fix. This does not at all appear to be trying to get the OP's company to pay for the repairs solely on the OP's behalf.