Ouch. Given that directive, the "best" solution would be to put in an
INTERNAL web server in front of the firewall, with a single hole poked
through it for PSQL access (TCP Port 1583 between the two computers).
You can then do all of your web development locally on the web server
there, first.

The advantage here is that initial costs are low, especially during
application development.  You can leverage easy access to the production
data, and you can easily build a test database area on the same
Pervasive database for your initial development efforts for testing
database updates and such.  Obviously, the dev process will take some
time, so it will be much easier with the DB in house.

Once the new app dev project is finished (OK, so programs are never
finished, just abandoned), then you can migrate it to an off-site
hosting facility if you really need to.  You probably won't find one
that supports native PSQL access, so you'll either need a co-lo facility
or one that at least gives you a dedicated (probably virtual) machine
onto which you can install PSQL.

Personally, my own recommendation would be to simply add the
Internet/Intranet license for your web users to the database engine and
keep everything in-house.  The cost of the extra bandwidth will be far
less than any co-lo-fac, although attaining a given SLA might be more
costly if they need guaranteed uptime.
    BB