Hi George
1. If you want to upgrade your 14.3 version, you have to follow the steps to actually get it upgraded from 14.3, which includes going through 14.4, 15.1, 15.2 and 15.3. This process also upgrades the database.
2. If you only do a new install on the new box, you will not be able to upgrade your database from 14.3 to 15.3, and connecting it will not work.
If you still want to have a new install of PPM application on the new boxes though, you could do both approach 1 and 2, meaning:
1. You upgrade your existing environments in place, including database from 14.3 to 15.3
2. You perform a new install of 15.3 on the new servers (direct new install)
3. You link the database to new servers and you restart
At this point I don't see a benefit to go via 15.2, unless there is something that you haven't mentioned.
Hope this helps -Nika