I believe the reason for the spot on the webpage to submit the quote is if you are trying to obtain an advance on the repatriation allowance. Otherwise you can submit the final invoice later - which actually works better since you'll have the final amount not an estimate. As mentioned it doesn't really matter which company it is as long as it is a legitimate invoice and within the allowable limit, which varies depending on where you're repatriating to (the entitlements per region are in the Staff Travel Manual, not the Employee Handbook).
I had the full amount reimbursed on my final pay, no issue.