I am not an expert on Celtic etymology, but I know that the origin of "paturnia" from Latin "pati" is not proven, and I also know that many Northern Italian words have Celtic origins, so I would not reckon a relationship between "bodhairim" and "paturnia" as totally impossible.

Most of the Italian etymologies I looked at link Italian paturnia (or paturna) 'sadness, melancholy; bad mood' to a Greek loanword from παθος (pathos) 'incident (good or bad), accident' < the verb πασχω 'to suffer'; the Greek word is related to Latin patior 'to suffer'.

As for northern dialects of Italian, I find a few references to Milanese patocch adj., paturgna n. (link), but also to Neapolitan a patùrniè (link). So, I'm not sure it's just northern Italian.

Though, the word is not recorded in Latin, the suffix -urn- does exist: e.g., diurnus 'by the day, of the day' from dies 'day', alburnus 'a kind of white fish' from albus 'white', eburnus 'of ivory' < ebur 'ivory'. In fact the melancholy Roman God Saturnus shares this suffix.

As for the Celtic languages spoken in the Italian peninsula, I would not choose Irish. There were several monasteries in Northern Italy founded by Irish monks, Bobbio comes to mind, but there were other non-Goidelic Celtic languages that are poorly attested, that seem more likely to have been Continental (Gaulish). In fact, some scholars see Lepontic as a Gaulish dialect spoken in Rhaetia and the Cisalpine region. It's really a question of when the word came into Italian and whether it came from Medieval monks speaking Old Irish or some pre-Italic (and possibly Celtic language) of Northern Italy. (And not all pre-Italic languages in Northern Italy were Celtic. Ligurian, for example, is not even thought by some to have been an Indo-European language. Old Irish bodar 'deaf' has a Welsh cognate [i[byddar[/i]. I don't quite follow how Old Irish went to Italian and then was borrowed into English. Or are you saying that Old Irish provided loanwords to both Italian (bodar > paturnia > pother) and English (bodar > bother)?

You would have to study the history of paturnia in Italian. I would also look at a Medieval Latin dictionary (Du Cange) to see if there was a Medieval Latin word that was similar.

Ceci n'est pas un seing.