Pricing at specific date.

I have a odd one, client signed proposal... (say 10k)

then asked to add another battery..... I verbally quoted a single battery cost. (say 2.5k)

I added the battery on OpenSolar and the cost therefor went up. (12.5k)

Final bill and the client claims the original quote, which he signed for, (pre the extra battery) was £1500 less. (8.5k) so the bill should be 11k.

Is there a method to show the changing in prices as the designs happened, or even just the total figure on the date he signed for the contract?