I see your position. This is how I would envision that analysis approach: Your dataset would have one observation for each farm, and so there would be no need for a RANDOM statement. Each observation would have values for Success (number of calves with FPT=1), Total (number of calves on which FPT was observed), pX1, pX2, pX3, pX4. Although different farms have different denominators for the pX, there is no uncertainty in these measurements (as long as we assume the farmer counts correctly) so measurement error models would not seem to apply here. Overdispersion is a possibility. Plus with only 41 calves, with 6 to 20 calves per farm, there must be very few farms (4? 5?), and you would be able to assess only one pXi at a time.
From another position, we could see herd success as being cumulative calf success. A model using calf-level data--where FPT is matched with X1, X2, X3, and X4 by calf--seems preferable. This approach would (1) avoid the concern about different farm sizes; any sources of variance among farms would be captured in the random effects for farm; (2) make overfitting (too many parameter estimates relative to the number of observations) somewhat less of a concern, or at least more manageable; (3) avoid the potential of overdispersion; and (4) better reflect the sampling design, where all variables are observed at the calf-level.
So, that's my opinion. From the point of view of continuing statistical education (a lifetime process), you could fit both models and compare what was possible and what story each tells. I'm certainly not suggesting that you then use the one that produces results that you prefer! But there is value (at least for me) in doing in addition to thinking.
Good luck!
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