It is worth noting that OP's original question before I edited it asked, "why would economists lie to us?" This already leaves a poor taste in my mouth; such a question is loaded enough as it is, only good for picking fights. The author stated at the bottom of the Progressive Dairy article is a lawyer, not an economist, and seems to be the basis for some of your figures in your analysis. The article from the Texas A&M AgriLife study cited by the Farm Bureau/NMPF did have economists writing on the topic, which I'll discuss briefly later.
Related to denesp's comment on cost of labor share, consider the firm's cost of milk production:
$$p_L q_L + \sum^n_{i = 1} p_iq_i = C$$
The cost $C$ is made up of the cost of labor (price of labor times quantity of labor) plus the cost of all other factors of production. The labor share of cost that was mentioned by OP would be:
$$\frac{p_L q_L}{C} \equiv \boxed{s_L = \frac{p_L q_L}{p_L q_L + \sum^n_{i = 1} p_iq_i}}$$
Suppose as in the hypothetical that the cost of labor doubles. OP argues that then,
we can multiply the labor share of a dairy farm's expenses per hundredweight by 2
but this is clearly not true from a mathematical standpoint.
$$2s_L \neq \frac{(2p_L) q_L}{(2p_L) q_L + \sum^n_{i = 1} p_iq_i}$$
More than this, it also assumes there are no cross-price effects. That is, if the price of one good changes, it will not change the quantity demanded of other factors of production. If there is any sort of substitutability OR complementarity between labor and any other factor of production, this cannot hold. Empirically, low-skill/blue-collar labor is a complement to high skill labor and a substitute with capital.
All of this walks around a bigger issue. The article in question makes a claim about the effect of disemployment on doubling milk prices. It does not say that it will come about by doubling labor prices, so the argument posed above is just a strawman. That is, it may very well be that doubling labor price would not double the final price of milk, but that is not what the article is arguing.
That said, I don't particularly like the analysis done in the Texas A&M AgriLife study cited either. Ultimately, it has no inference/causal modelling and is just a survey with some summary statistics, with a lot of questionable assumptions. There is like one price elasticity in that whole analysis as I could see, and its use was not compelling.
The lazy way to interpret this is to just say the methods of the study are bad and that its conclusions have no value and only serve to mislead the public. I don't think this is particularly true either. Private sector market analysis is definitely in some ways less stringent than publishing in, say, an economics journal however.
So what can we do?
The long, boring, rigorous way to estimate the impact of a large shock to the labor supply is to look at labor demand and labor supply simultaneously.
Consider an extremely simple set of linear estimations:
$$\ln(L_s) = \alpha_s \ln(w) + \vec{\beta}_s \vec{Z} + \epsilon_s$$
$$\ln(L_d) = \alpha_d \ln(w) + \vec{\beta}_d \vec{X} + \epsilon_l$$
Where $Z$ is a vector of control variables/supply shifters (e.g. non-labor income), and $X$ is the same but for demand. Notice that labor, $L$ and wage, $w$ are endogenously determined, which nudges us towards the use of IV estimation, rather than the typical OLS.
Labor Demand:
From here you'd have to think about what kind of functional form production should take: $Y = f(K, L, \cdots)$, where $K$ is capital, and you can put in some other factors if you want (legal labor vs illegal labor perhaps?), and derive conditional factor demands, which you can plug into the equation for cost which gets some $C(w, r, \cdots, Y)$, with $r$ representing the rental rate of capital.
Applying Shephard's Lemma, you can then derive the elasticities of substitution between two different factors of production, which are used to calculate the cost shares of of each production factor, and eventually, we can then find the own-price elasticity of the factors, used to help estimate $\alpha_d$.
Labor Supply:
Similar line of thought, but with a specification of utility and working with Frisch elasticities.
Plugging in the data (number of laborers, capital stock on farms, etc.) could prove difficult or impossible without existing data of course.
This is obviously a very rough sketch for an empirical framework for analyzing labor supply, but it is a foundation for analyzing all sorts of shocks to the labor supply. For a more detailed analysis, you can refer to Cahuc, Carcillo, and Zylberberg and their first few chapters on labor supply and labor demand.
Sometimes in the economic literature, undertaking a more detailed study like this requires a lot of funding and time, with in mind particularly the hassle of data. People want practical policy answers in the short run, particularly in the private sector. With limited resources, sometimes studies like the ones you've looked at provide "second-best results" and that is the analysis we have to work with. The productive, honest thing to do with studies that take on partisan issues is to think about how to feasibly improve them and ask specifically about how to look at the research being done on the topic.
There are plenty of voices accusing all manner of partisan research/analysis as misleading, and still plenty polemics out there who would waste their time writing polemics. It is not for this site.
