If the centre is 2 hours away I would drive there when it opens and check, or at the very least get in touch with the manager that day of that office - not the UK helpline.
Next I suggest explaining the situation with a letter in your ILR application. I dont think you need the physical BRP for the ILR application, but where you might get stuffed is if you lose it, don't report it and don't re-apply within 3 months of losing it. Being open and honest is vital. Also if someone from the centre has found it they are supposed to return it to the address below, presuming the LITUK people follow procedure...
There is some clear guidance if a BRP is lost - and the Home Office considers it sufficiently important they will get in touch within 1 day of reporting it.
https://www.gov.uk/biometric-residence-permits/lost-stolen-damaged#:~:text=or%20stolen%20BRC.-,If%20you're%20in%20the%20UK,a%20replacement%20within%203%20months.
If your BRP is lost or stolenYou can report your biometric residence permit (BRP) lost or stolen from inside or outside the UK. You can only apply for a replacement from inside the UK.
The Home Office will contact you within one working day of reporting it.
You can get someone to report for you, for example a legal representative, a charity, employer, college or university.
There are other ways to prove your immigration status while you wait for a replacement BRP.
BRPs are different from biometric residence cards (BRCs). BRCs are also called ‘UK residence cards’. You might have one if you’re in the UK as a family member of someone from the EU, Switzerland, Norway, Iceland or Liechtenstein. Find out how to report a lost or stolen BRC.
If you’re in the UKIf your lost or stolen BRP was valid for 3 months or more, report it and apply for a replacement. You must do this within 3 months of losing it.
You can be fined up to £1,000 and made to leave the UK if you do not apply for a replacement within 3 months.
If your BRP was valid for 3 months or less, you must do one of the following:
report it as lost or stolen if you do not intend to remain in the UK after its expiry date
apply for a replacement if you plan to leave and re-enter the UK within 3 months of its expiry date
apply to extend your visa if you want to stay in the UK after its expiry date - if granted, you’ll automatically get a new BRP
If you find your BRP after reporting it lost or stolenYou will not be able to use your BRP if you find it after you report it lost or stolen.
If you’re in the UK, you’ll still need to apply for a replacement.
If you’re outside the UK, you’ll still need to apply for a ‘replacement BRP visa’ to travel back to the UK before applying for a replacement.
When your replacement arrives, cut your older BRP into pieces and post it to the Home Office. Make sure you cut through:
your personal details
the royal coat of arms
the gold biometric chip, if your card has one
Returns Unit
PO Box 195
Bristol
BS20 1BT
If you’ve found someone else’s lost or stolen BRPIf you’re in the UK, post the BRP to the Home Office using an envelope without a window.
Returns Unit
PO Box 195
Bristol
BS20 1BT
Do not send any other items or documents with the BRP - they’ll be destroyed.
If you’re outside the UK, hand the BRP in to the police or to the nearest British embassy or consulate.
Good luck!...