I think the message suggests that you go to the Lithuanian Archives! SVaiginai is well to the north end of the Polish border of Suwalki gubernia, well away from the present day border with Poland. So my own guess is that the records are in the Historical Archives of Lithuania rather than in the Polish Archives if they are over 100 years old. More recent records may well be held by local civil authorities rather than the Archives of Poland.
If I were you I'd send them an email with message from the Polish Archives and the information you need (given and surnames with variant spellings; approximate year of birth; and name of birthplace or parish of baptism) and ask them to look for the baptismal record. If they then respond that the records are in Poland, you can use their message to the Polish Archives as evidence of your claim. And if they have the record (and it has survived), they'll let you know how to obtain a translation version of it. Here is their web site in English with email address: