A churn is cuinneog and a butter-making churn is cuinneog ime, but how do you express the idea of the volume of what is contained in a churn - as in this case - butter ?
A teaspoonful is - lán taespúnóige whereas a table spoonful is given in the dictionary as simply spúnóg bhoird.
This is taken from Darby O'Gill and the Good People. I thought that I might find some vocabulary about butter-making on the website of the Cork Butter Museum but it's all in English.

Lán cuinneoige d'im, b'fhéidir?

b'fhéidir

or maybe:

cuinneog lán d'im
churn full of butter
or
cuinneog ime
churn of butter
[this might just mean "butter churn", as you said, but it might have both meanings]

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

This is more of a musing than offering an explanation..

I would see cuinneog lán d'im as a churn full of butter e.g. Tá an chuinneog lán d'im The churn is full of butter
and lán cuinneoige d'im as a churnful of butter e.g. Tá lán cuinneoige d'im sa bhuicéad A churnful of butter is in the bucket

Would that make sense?

Here's the context (taken from Darby O'Gill and the Good People - collected by Herminie Templeton Kavanagh) -
Right and left, generation after generation, the fairies had stolen pigs, young childher, old women, young men, cows, churnings of butter from other people, but had never bothered any of our kith or kin until, for some mysterious rayson, they soured on Darby, and took the eldest of his three foine pigs.