Can I Update Time Machine Backup Without Booting From The Source Drive

Greetings,

Can I update a time machine backup without booting from the original source drive?

My main Macbook Pro internal drive seemed to have died. I had not backed up in months so my Time Machine backup was old.

I replaced the drive with a new one and settled for restoring from the old TM backup because my latest info was on the dead drive and not backed up.

-----------

I just plugged the "broken" drive into a dock and it mounted.

It won't boot from the drive but I can mount it and copy from it so far.

I now want to update my Time Machine backup with the info on the 'broken" drive while it's able to be mounted.

How can I accomplish this without booting from the 'broken" drive and running time machine.

Recap...

I need to update a Time Machine backup using another hard drive without booting from the source (as I cannot). Both drives are connected via USB to my MBP and I am booted from another MBP. I want to end up with the newer info from the "broken" drive updating my older Time Machine backup.

Thank You,

SG