Third-party car insurance claim in PH - insurer approved ₱33.5k but casa estimate is ₱83k. Who pays the difference?

I was involved in a minor car accident where I was at fault, and my insurance (Standard Insurance) is processing the third-party claim.

The third-party vehicle was brought to an authorized dealership. The dealership’s estimate is around ₱83,000, but Standard’s current LOA/RVR only approves ₱33,567.41.

The biggest difference appears to be a left headlamp assembly costing around ₱45,000. The RVR says to remove and reinstall the LH headlamp to allow repairs, but the headlamp assembly itself isn’t included among the approved replacement parts. The RVR also says additional hidden accident-related damage can be submitted for reinspection and approval.

The third party is also asking me to arrange and pay for an automatic rental car for her day-to-day transportation while her car is being repaired.
I’m currently asking the insurer whether the rental is covered and how the additional repair items should be handled.

For those who have experienced a similar situation:
- If the casa estimate is much higher than the insurer’s LOA, how was the difference handled?
- If a part is found to be damaged during dismantling, does the repair shop normally request supplemental approval from the insurer?
- If the insurer doesn’t cover a rental/replacement vehicle, does the at-fault party normally have to arrange one personally?
- Should I avoid agreeing to or paying for anything outside the insurer’s approved claim until Standard confirms it?

I’d appreciate experiences from anyone who’s actually dealt with a similar third-party claim in the Philippines.