In checking the state archives in Lomza, it shows that they do not have Roman Catholic records for the year range: http://baza.archiwa.gov.pl/sezam/pradziad.eng.php
If the church has the record, your cousin can find it. The books should be indexed. All she/he has to do it view the index. She will need to know how the name is spelled in Russian. Once she finds it in the index, she needs to go to the record. "Normally" everything in the documents is in Russian, but they also have the names of the parents and child in Polish. The books from my grandparents parishes were recorded in this manner. Surely she can find someone, who still remembers Russian to go there and help her. If not, let me know. I have the "old" Russian alphabet, which I use. Before viewing microfilm, I learn how to write the surnames in Russian so that I can recognize them in the indexes. Let me know if you would like to have a copy of it. This used to be on the Internet in pdf. I saved it many years ago.
The archives has Lutheran, Jewish, Protestant and Roman Catholic record. No civil records are listed.