I guess I don't see why local farmers _wouldn't_ raise their prices to maintain their premium above store-bought eggs. They certainly have around here.
Because in setups like I'm describing the farmers tend to start building relationships with their customers. Price gouging is generally not the best way to maintain those relationships.
In previous egg shortages over the years the couple of farmers I use would sometimes impose limits like 1 carton per customer or something like that. But not jack up prices.
Smaller farmers usually want to make money to cover their costs, and if they weren't doing that, they're already in a huge pickle.
As you mention, they'll impose limits (or perhaps offer "shiny brown eggs for $1 more") rather than piss off the customers, who 90% of the time have cheaper options already.
Maybe they see it as a way to acquire new customers. They are in the very unusual position of being able to undercut their bigger competitors on price while still making a profit.
Right now, buyers are probably shopping around a ton. You can probably get customers who normally wouldn't be interested. After they try it, some of them may decide they like it and could become long term customers.