Upgrade to WP 3.8 safely from 3.4.1 without losing customisation

To contradict the previous developer, it is unwise not to upgrade.

Some theme customisations may require a bit of tweaking: that depends what exactly those are – e.g. CSS/HTML customisations shouldn’t be much of an issue, since they should by and large be WordPress-agnostic. If you’re using out-of-date plugins, these might have to be replaced with new ones. But I would regard any such obstacles as minor compared to the possible security vulnerabilities and reduced extensibility and customisability that result from not upgrading.

The solution should never be “we’ll keep out-of-date software in order to preserve our customisations”. The solution should be “we’ll upgrade the software progressively and make the modifications necessary to ensure that everything continues to work smoothly”.

You could create a test installation of WordPress somewhere, using 3.8. That will give you a sandbox with which you can test your theme, upgrade to the latest versions of all plugins, and find fixes for what isn’t working. You can download your current theme via FTP from the /wp-content/themes directory, and upload it to your test site. (You could also copy your database over to simulate the live environment if you wanted.)

You can install WP on localhost (using WAMP or XAMPP) or on any server.

There are plugins available that will allow you to backup WordPress in whole (if you want everything) or in part (if, for instance, you just want certain assets like the theme) directly through the WP back end. I can’t personally vouch for any particular solution, but if you do a search for backup plugins in the plugins directory you should find something that suits your needs: e.g. this one.