As you are in the UK suggest you shove this one straight up to CHIRP.
From what you say the company are asking you to do the impossible - but why? Doesn't normally take that long to sort through ones own admin stuff before getting on with the flight planning etc. Could you meet their requirements by reviewing the admin prior to check out of your previous flight?
Are you able to access the computer from your home? Sounds
odd but several companies now do this.
If it is just a blatant attempt to get you to come in early then they are, I think, in breach of the intent of flight duty times and limitations.
Check with the authorities, I suggest.