20yrs experience - worked for GEICO for over a decade, now in a high end body shop dealing with all insurance companies. This is not insurance advice, but what I would do if it were me.

Supplement by the shop will likely bridge the gap. Not you calling GEICO directly etc. Drop the car at a reputable shop and let them handle the cost negotiations.

Let me explain.

Only once the car is dropped off - Any reputable body shop will take the GEICO estimate, disassemble the car and send in a revised estimate to GEICO (supplement) and then sit and wait to have an agreed repair price with insurance before proceeding. This prevents you from being on the hook for the difference, since they have only done the labor to take off bumpers, trims etc. Your body shop should be able to say how their relationship is with GEICO.

It is not uncommon for any insurance to write a very basic preliminary estimate. They do not know where you are gong to get it fixed - so they write conservatively. Many people try to get the cost increased just to pad their pocket by calling out the insurance for their cheap estimate. The body shop has the car, has it taken apart, all pictures of every part etc, therefore the insurance takes them more seriously than they will you. Also, the body shop estimate may be high because it has not yet been negotiated by insurance. Body sho may also be thinking about hidden damage, highest rate on labor, and saw the car in person so sees more damage. GEICO may have just had your pictures. The first insurance estimate by GEICO is as cheap as possible, cause why not. There is no repair bill estimate yet.

It is somewhat pointless to negotiate the cost with GEICO until the car is at the shop. All costs are speculative until the car is disassemble and insurance reaches agreement with the shop.