With 2K GBP, you can just book a J RT from LHR to US? They're literally 1500 GBP right now for LHR-NYC, 2200-2400 GBP to SFO/LAX. I would just buy my own tickets. The company is being very reasonable here --> your HR person may not have the authority to change the travel policy, but they are making it up to you with straight cash.
I also echo other sentiments on what other offers you have. If I had a job offer that was 2k less per month but had J travel, I would definitely take the cash option and spend to upgrade whenever I wanted.