You have to buy licenses for each computer. THe 5 user license is not licensed for 5 computers but allows 5 users to connect to the engine. You'll need a 5 user license on each machine unless you can get a single user license (I'm not sure what's available from a license key standpoint).
Which company did you talk to? I would suggest contacting Pervasive directly (1-800-287-4383 in the US). They should be able to straighten you out.
One more item of note, the PSQL v10 engine on the laptop may have problems connecting to the server if it's running PSQL v8. Ideally you would upgrade all clients and the server to PSQL V10.