Can internal supporting documentation such as the IGE, negotiation memoranda, or Congressional Notification essentially place a cap on how much can be spent and exercised under the contract? Can the IGE (or other) take precedence over a negotiated, binding contract?
The contract ceiling is what is on contract. The IGE has no effect on the contract ceiling after award; it was established to help the Government get an idea of the budget they need to prepare and what they might expect from offerors. Since we generally do not release the IGE, it cannot be binding on a contractor who is unaware of it.
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The Government is bound to the dollars obligated on contract. If $250M are on contract, that is the obligation; if $283M are on contract, that is the obligation. Assuming your source selection team evaluated all options, you can exercise each option and obligate the value of the option. The only limit you have is the negotiated value of the option(s) and the availability of funds. IGE was an estimate, and the Congressional Notification was of the instant value of the contract.
If one were to follow the rationale used in the Question Background "....because the Independent Government Estimate, the Congressional Notification, and other supporting documentation only supports $283M...." then you might be prevented from modifications should the dollar value of the next mod exceed the IGE developed years ago under different circumstances.