When driving in Belgium, what equipment is mandatory to have in a car registered in another country?

They are not mandatory for foreign cars.
My parents live in Belgium and I live just across the border in the Netherlands.
I visit my parents almost daily, by car, so I had reason to check this myself.

I checked the Belgian traffic code:
This indicates clearly that the extinguisher and the first-aid kit are considered to be "part of the vehicle" and therefore only apply to vehicles that have a Belgian registration. (They must have the kit in order to be considered road-legal.) This is not enforceable on foreign cars: Because of international traffic regulations Belgium is required to allow road-legal cars from other nations on their roads as long as they are reasonably safe to drive. The presence (or absence) of an add-on not essential to the operation of the car (like a first-aid kit) can't be part of any restriction that prohibits foreign cars on Belgian roads.
The triangular sign and the vest are mandatory to use in case of an accident, car-trouble or other dangerous situation. If you don't have them in the car you are not in trouble, unless you get in a situation where you are required to actually use them.
If you don't have a vest and need to leave the car on the middle lane of a 3-lane motorway the policy-officer that shows up, will fine you for not having one. (Happened to my cousin when his car got in a fender-bender on the Brussels ring-road.)

I also asked my parents neighbor who is a Belgian police officer:
He confirmed everything above too. He also mentioned that the fire-extinguisher needs to be a certified model and not beyond its expiration date either.
This is effectively never checked, unless the police officer is really looking for a reason to hassle you.
(He hazarded a guess that at least 90% of Belgians drive around with an expired, or empty, extinguisher. The thing is only in the car to satisfy a cursory glance of the police.)

The Dutch and British automobile clubs both suggest that fire extinguishers and first-aid kits are only compulsory for cars registered locally as do the ACL from Luxembourg. Not necessarily authoritative but I think they would know.

For some reason, the “service public fédéral mobilité” (that's the name of the federal ministry for transport) has nothing to say about this and only mentions safety vests, which is not particularly helpful…

It seems that the basis for all these rules is the règlement technique des véhicules automobiles (in particular articles 70 and 71). Article 2 provides that

Sont soumis aux prescriptions du présent règlement général, les véhicules automobiles circulant sous couvert d'une plaque d'immatriculation belge, ainsi que les remorques belges tirées par eux.

This would seem to be authoritative and to confirm that foreign-registered cars are exempt from all these requirements (but it's always dangerous to try to interpret a legal text when you know nothing about the legal system…).

A first-aid kit is quite cheap and recommended or mandatory in many places so you could probably consider getting one even if a fire extinguisher seems too much of a hassle.

Finally, and keeping in mind that such anecdotes are not worth much, I have had many occasions to drive in Belgium and have never experienced or witnessed a police check on the road so I would not worry too much about it.

Although the Belgian road code does mention triangle, estinguisher, first-aid kit and high-visibility jacket (see articles 70-70bis-71 here), according to the Bison Fute governmental website, you must have:

• 1x Triangle
• 1x high-visibility vest per passenger
• 1x Fire extinguisher

Quoting from the website:

Equipement obligatoire à bord du véhicule

• 1 triangle de signalisation
• 1 gilet par occupant du véhicule
• 1 extincteur

This seems to agree with this other French website on the topic. Nowhere does either website mention a first-aid kit.

To be extra sure I would follow the advice on the Bison Fute website, and I would ask the first police officer you meet:

Ces informations sont données à titre indicatif et ne sont pas exhaustives. Il est conseillé de s’informer au passage à la frontière sur les autres réglementations en vigueur.