SQL Server replicating data to an archive
if i understand correctly log shipping and transactional replication is not an option because they basically update on the other side whatever you do on the primary server.
you will get an updated database and if the info is overridden so is the secondary database.
i would maybe try create a planned DWH database that i would copy the fresh data every period of time in some way.. maybe SSIS or bulk insert.

looks like it'll take some work like understanding when is data being overridden: every 5 minutes? at random times when they update it in the app? and then decide accordingly on what to do(maybe even use a trigger on insert or update to bulk insert it to a DWH table in s different server).
I hope this helps.
keep in mind if the updates and insert happen often this triggers may cause performance issues in some cases if done constantly.