But the Syriac Peshitta, as we have it today, is not written in 2nd century Syriac. It's written in 5th century Syriac (the n- preformative and a bunch of other features gives that away). Even the Old Syriac Gospels (which aren't written in Old Syriac – the Syriac dialect contemporary to Jesus – but in early Classical Syriac) are still too young.
Sadly, there are no NT texts that survive in Galilean, outside of a number of phrases and underlying puns in the Greek NT, but there is a fragmentary Christian Palestinian Aramaic NT which is written in a dialect closely related to Galilean. It, however, is very obviously a translation from the Greek (e.g. it refers to Jesus as "Yesus" instead of Yeshua, John as "Yohanes" instead of Yohanan, etc. – all of the names are transliterated from the Greek) and was made somewhere between the 4th and 5th centuries. (It's contemporary with the Peshitta.)