How to, efficiently, search through many sql dump files of really big size
ApexSQL Restore will mount the backup as virtual Database, and you can query it using SSMS.

I also published a PowerShell script that will let you analyze your backups in a given folder. I wrote it for analyzing size but you could easily adjust this to include properties like the date taken and such to find which backup you need to mount. You can find the script and article around it here. 

I have not played with the Apex tool extensively but do not believe it will search all backups in a folder for you to find the one with the specific data you need. You would have to mount at most the full and then all the logs to get it to the date you need. 

I have never seen one that does exactly what you are looking for, but the above should help I think.