ILR requirement for job

Can jobs list ILR as an essential requirement? My wife and I are 4 months away from being eligible for ILR. I came over on the old 5 year work permit so I didn't need to apply for FLR. If we currently have the right to work in the UK, and expect to continue to have that right, can we really be rejected?

Your work permit is only valid for your current employer. So if you are the work permit holder and you want to change jobs, then the new employer would need to sponsor you. If you are asking about your dependent who has the right to work, then he or she can work for anyone. Basically it sounds like the company is saying they won't sponsor a visa.

I am asking about my dependent so they wouldn't have to sponsor anyone. The requirements say "EU passport or Tier 2 visa with unlimited leave to remain"

The job shouldn't require any type of security clearance and isn't for the government. It will require up to 80 days of international travel a year. We are close enough to ILR that that shouldn't matter.

They can't place you on a contract that would go past the visa expiry date, the fact you will apply for ILR and almost certainly get it is irrelevant, to them you may not. So they would only be able to give a temp contract.
My wife got a job after 1 year here, so another year to ILR, they put her on a short term contract but with an agreement this was extendable. When it was time for her ILR they requested proof that she had applied (she gave them a copy of the letter to say the application had been received), so that was acceptable to extend the contract but they needed proof of the ILR as soon as it was received.
It may also depend on the size of the company - this is a big bank, so they actually have a department within HR that deals specifically with foreign nationals being employed so they are on the ball with regards immigration legislation.