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If you have price data going back several years, but the price is constant, is there any way to work out the elasticity of demand (or dynamically optimise prices another way).
The sales volume is in the tens per day, so not tons of data for A/B testing.
I also am trying to avoid limiting the data to competitor pricing/industry pricing.
I have a variety of information about the customers who are buying the product. I also have multiple SKUs to compare.
No, to estimate elasticity you need at least some variation in prices. When you estimate the coefficient for elasticity using any parametric model, the estimate will depend on variance of prices, for example using regression the parameter would be given by $\beta = \frac{cov(p,q)}{var(p)}$ where $q$ are sales and $p$ price. If $p$ is fixed then var(p)=0 and division by zero is not defined. You should ask whoever can set the price to run at least some experiments with trying to vary the price.
In addition, note price and sales volume is not enough to figure out what elasticity is even if price is not constant. Price and quantity (sales) are endogenously determined. That means whenever you want to estimate price elasticity of demand you cannot do that from observed price and sales data alone (such data are just collection of equilibrium points between supply and demand). You need to have some extra data on some instrument that captures the supply side (e.g. firm costs) and run some simultaneous equations model.
