I see Van Hool logo, maybe FIAT is a chassis?

You are right. The question is: what element determines the "marque" of a bus? I do not insist on FIAT.

Judging by only its looks and a quick search, it looks like a Van Hool 320 on a Fiat chassis. But I can't say for sure

In the scope of this website, "model" (or "marque") is an equivalent of vehicle body manufacturer. It does not have to be "only Van Hool" or "only FIAT".

Based on the pattern that I can see from your priceless pictures, Van Hool used to produce bodies that were put on various FIAT chassis. Main question is - what do 3 digit numbers represent: Van Hool body name, FIAT chassis type or both.
306 - https://fotobus.me/photo/3914328/ - kind of fits FIAT 306 chassis and Wikipedia article.
409 - https://fotobus.me/vehicle/2548166 - same text style but could be both for Van Hool and FIAT
700 - https://fotobus.me/vehicle/2548166 - 700S28 seems to be an extension of FIAT (same font) but there is very similar bus that is definitely "Van Hool 700" according to manufacturer's plate https://fotobus.me/vehicle/2191796/#n3103617

https://en.wikipedia.org/wiki/Van_Hool:
On February 15, 1957, Van Hool signed a commercial agreement with Fiat; Van Hool would incorporate Fiat engines and other mechanical components (gearboxes, axles, steering) in its vehicles. It developed from a coachbuilder to a Belgian manufacturer of integral buses and coaches, known as Van Hool-Fiat.

Given that the Van Hool 700 has only a Van Hool plate and no FIAT plate (and only a Van Hool "VIN", with no FIAT number), it can be assumed that a complete chassis was not actually used, only components.