You are a party to the contract. This means you have to protect your party's interests in the agreement, which you are NOT doing now. You say your customer does not want to pay for a fully loaded team but wants the benefits of a fully loaded team. All customers would want that. You answer needs to be: pound sand (not literally).
Your price for a small change request is the greater of the actual effort required or the cost of the resource that you are unable to sell elsewhere. The net result would be a fully loaded team for that change request. Here's an example: if a team had to travel for two whole days round trip in order to do work that takes only four hours, do you believe the customer would only pay for the four hours? No way. They would be on the hook to pay for the four hours plus the 16 for travel.
If you are concerned you will lose the business to an alternative, if there's an alternative willing to lose money, let them. No contract is worth it if you are not meeting your interests in the agreement. There are other clients out there; go get them.
Your FP formula is: Your costs + Your margin + risk = fixed price.
Your costs include unused utilization of your resources. It is part of the cost burden.
EDIT to answer question in comment: You indicated your original scope at your original FP is finished so there is nothing more to do with that. Whatever CRs you have already agreed to at whatever price you indicated is a commitment you cannot back out of. You need to complete that scope and invoice for that price and live with the reduce profit or loss you experience. For future CRs not yet agreed to, here is your chance to price it building in the extra cost and margin for non-allocated hours for which your resources are not fully loaded. It does not matter if it is T&M or FP or CP. What matters is you load your total costs in your price build up.
Since this will show an increase in price for your customer, you need to have a sit down and explain the deal. Since you are a party to the contract, you have the right to explain your position in the deal. If your customer balks, then let him compete it. Every other seller will build the price the same way.