If you want to renegotiate your price you should do it before the contract is signed. Most inspection contracts have the fee as well as what is included and not included for that fee.
However I would not ever renegotiate my fee upward on site unless there was something that not disclosed properly to me during the prescreen wand even then it would have to be a very substantial discrepancy.
When there are many many issues with a particular system instead of listing them all I will simply state that here are some of the deficiencies but there are many to list all of them and recommend that a specialist review the entire system.
What if the client does not agree with the revised fee? Are they now under undue stress to pay it anyway because of the time limitations in their purchase agreement? How would a court of law look at it? If the client came from an real estate agent referral, would you expect to be getting any future referrals from that agent?
Sometimes the best business decision can be just to bite the bullet.