How can I copy a working Mac Os X Tiger 10.4 install to my new HDD without destroying the Mac Os 9 installed?
Oh goody a classic MacOS question...

The nice thing about the classic MacOS is that once it is installed you need little else besides the files in the system folder. Yes there is the "blessing" of the system folder but if it is your intention to use it in a "Classic Environment" inside macOS then you should not need to worry about that.

Connect the drive with the macOS Tiger installation and boot from that. I think that will work. Some older Macs cant boot from USB. Failing that can you put it in an old firewire enclosure?

The goal here is that once booted into macOS Tiger just drag the folders from the classic install on the SSD to your Tiger drive. Put them at the root of your (Tiger) Mac HD and take the whole system folder, applications folder and anything else you need.

I did this a while ago and while I was at it I made a zip archive of the whole thing, "just in case." Just in case happened and I unzipped the files and it worked a treat.

Once you have done that you can use an old (shareware) copy of Carbon Copy Cloner to "back up" the Tiger drive with MacOS 9 on it to the internal SSD.