If the church is one of the closed ones, you should be able to get a copy from the Archdiocese of Detroit (AOD).

http://www.aod.org/parishes/sacramental-records/find-your-re...

Is the baptismal dated 1930 or earlier? If not, you are not supposed to copy the records for baptisms at the BHC.

I was told about a year or so ago, that if they have the time, one of the BHC staff may be able to look for the record if you have a date and location. Did you try contacting the Burton about that?

This is what I got when I contacted the AOD about a year ago.


It is the policy of the Archdiocese of Detroit that sacramental records
may be used for research by parish staff if the records were created
prior to 1930. Researchers may attempt to access these records in one of
two ways: through the sacramental microfilms at the Burton Historical
Collection or by contacting the parish directly for research assistance.
Parishes may do research for people in these records; however, they are
not obligated to do so. Due to preservation and privacy concerns the
public is not permitted to handle the original sacramental records from
a parish. The records at the Burton Historical Collection are copies of
the original sacramental registers which are maintained by the parish.
Since the records from which you are requesting information are
approximately one hundred and fifty years old the legibility of the
original sacramental registers is likely not much better than the
microfilms available at the Burton Historical Collection. These records
are hand written and the originals are often extremely difficult to
read.

The Archdiocese of Detroit Archives cannot compel a parish to do
research for genealogy in their sacramental records. Many parishes do
not have the available staff to provide research services, which is why
the sacramental microfilms have been made available to the public at the
Burton Historical Collection.