Your best option would be to make sure the appraiser had observed all improvements. You will never get them to change by talking with them. Not in todays world.
But if the many upgrades and rehab were not detailed they may have missed them. Example: New HVAC system. If appraiser doesn't know they cannot use. Not full proof by any means but an angle that is non confrontational and actually helps the appraiser provide a more reliable product for the lender.
Other item could be the market has changed. In FL a 15% drop over 4 months is realistic. But a completely rehab'd property may not be represented by a trashed short sale. That can always be an option when you see the appraisal. The net and gross adjustments for comps have to bracketed and not be very large. Maybe 10-15% or they are not true comps. If you can assist the appraiser in making a better product that may help but don't expect anything. For 5K you may have to deal with it.
If it was 10k plus and you were confident you could pay for a 2nd independenyt appraisal but lender may still use the lower depending on the AVM review.
I have had 50/50 success with appraisers but its a headache and slow either way.
If they change anything they look incompetent and they don't like to do that. If they wer uninformed of something that affects value then you may have an option.