As it turns out, there is more information about the Project 4202 test record than I knew existed. A Twitter post drew my attention to a very detailed table of Soviet/Russian launches that Jonathan McDowell keeps on his planet4589.org site. The table has an unusual amount of detail in that it includes serial numbers of launchers. (Jonathan also maintains a master list of all launches, which includes those for which serial numbers may not be available.)
In the list, the Project 4202-related launches are designated 15A35P, after a modification of the 15A35 UR-100NUTTH/SS-19 missile. There are nine tests of this type in the table and in general the details correspond to the information collected from other sources (note that I corrected a couple of dates in that table). There are important additions as well, but there are interesting differences and omissions. Below [on the other side of the link. ed] is a combined table that shows all known and suspected tests related to the program.