I personally would:
1. create a back up of the current site and db.
2. upgrade it from 1.5 to 1.6ST in place.
3. see if it still works. If it does then continue. If not restore and try again correcting any issues.
4. create a back up of the new site and db.
5. export the db.
6. setup everything that needs to be setup on the new server.
this include empty db, user/pass, webserver etc.
7. restore the back up made in step 4.
8. install the 1.7.x code base, and run the update.