Parsing “have a limited release the product”

Although there is already an accepted answer, I will try to give what I think is a more complete account.

First of all, the sentence is definitely wrong as written. Although there are several ways to correct it, I agree with another answer that it is probably missing the word "of", and should be:

"We'll have a limited release of the product and let this region serve as a guinea pig."

(The following sentence in the original is helpful in interpreting it: "If it is received well, we can expand production and distribution to the rest of the county.")

In the original context, the sentence is being used to illustrate the idiom "guinea pig", referring to a small animal which is often used in scientific research: https://en.wikipedia.org/wiki/Guinea_pig#In_scientific_research . Idiomatically, "guinea pig" here means a test subject, on whom research is being performed.

The sentence as a whole is in the jargon of product marketing or sales. A "release" is the event of making a new product available to customers. (This is also used in related contexts, like programmers releasing software.) A "limited release" means that a new product (or a new version of an existing product) is being made available only to some customers -- in this example, to people in a certain geographical area, but it could also mean only for a certain time period. From the next sentence, we can guess what kind of area. (A "county" is a United States geographic subdivision smaller than a state/province, but larger than a city. Although this could be another error, for "country"; that error is common.)

So, "to have a limited release" means the same thing here as "to release in a limited way". The verb "to have" is serving a generic purpose here, "to have [some event]", meaning to make some event happen. You could also say "to do a limited release" or "to hold a limited release" (the same way you might say "to hold a party".)

Another choice would be to remove "the product"

We'll have a limited release and let this region serve as a guinea pig.

In this construction both what is being released and the region would have to be inferred from context.

We'll have a limited release the product and let this region serve as a guinea pig.

This is an error.

Notice that examples such as this are usually discovered by computer search of thousands of documents and not checked by humans unless someone complains. The program used for the search has no real-world knowledge and so does not routinely make corrections or sift out errors.

Some online dictionaries make the above advice explicit.

The smallest change to something that makes sense is to insert "of" giving

We'll have a limited release of the product and let this region serve as a guinea pig.

In this case "limited" is an adjective applying to the noun "release". Presumably a particular region is specified by the context, and will be used as a test site.