If by "properly" you mean resistant to attack then no Bitcoin won't function properly with only a few hundred nodes. Cancer nodes, DDOS, information hostaging, null routing, network fragmentation, feasible 0-confirm double spend all become realistic attack vectors if the number of nodes is small.
What do you mean by "bitcoin"? The bitcoin distributed consensus algorithm? The bitcoin currency formed by it? Or the current version of the client software we happen to be using today?
In a world where there were only a few hundred full nodes we wouldn't be using the current software— we'd be using some evolved version of it. The bitcoin distributed consensus algorithm and the bitcoin currency can function fine in that world, the current software as is— not so much.
The P2P protocol is simply not essential to bitcoin. Bitcoin can run over just about any transport, and there are already people communicating bitcoin across things which are not the P2P protocol. (e.g. compressed blockchain dumps, json messages, etc). As bitcoin grows and evolves many different transports will coexist (and no one could stop them from existing, even if we wanted to)
To make node software for the world with a few hundred nodes you'd want do to things like explicitly configure sets of 'known not-completely evil peers' with cryptographic authentication like the freenet darknet. This kills cancer-nodes, confines DDOS and hostaging, prevent null routing and fragmentation, etc. And its pretty easy to maintain in a world with a few hundred nodes. You can even fully mesh the main nodes to each other.
While I don't think "a few hundred nodes" is a desirable (or plausible) outcome, we're already getting some of the features needed for that hypothetical universe (see the 'keepnode' pull request) because the attacks you've described are not impossible enough for high value targets in our current diverse network but are made much more difficult with a little sprinkling of some pairwise semi-trust (because they're all mostly broken by 'have at least one honest peer'). No amount of P2P network diversity helps you if your ISP is evil and isolates your node onto a fake network... but a couple of authenticated peers elsewhere on the network kills that attack dead.