If all the sockets in the room (or several rooms) are connected to the same breaker, then yes - you need to consider all the equipment that you have plugged in there, and you may already be close to the 3600W limit.
What's the circuit breaker rated at? It's probably 16A, in which case I probably wouldn't bother getting a powerboard with a fuse or circuit breaker on it. As you stated, it'll be hard to get to. And the circuit-breaker in your fusebox will probably still blow first if your fridge is also on the circuit, and you'll still lose power to the fridge.

But electricians will often run extra circuits to locations where they expect washing machines, dryers, dishwashers, and stoves to be placed, because they know that these devices need more power. So you need to check that.
Your powerboard has a 16A limit. That applies to the plug, to each of the sockets, and to the board itself. None of these components should exceed the 16A limit. Plugging in 3 devices that need 5A is ok (the plug is carrying 15A). Plugging in devices that need 15A, 0.5A, and 0.5A is okay (the plug is carrying 16A).
But plugging in two devices that need 10A is a problem - you're loading the plug (and the cable, the wall socket, and your wiring) to 20A. Assuming that both devices are drawing their rated power at the same time, that is.
Your options are:
- don't run both at the same time.
- have an electrician put in a new socket on a new circuit.
- run an extension cable from elsewhere in the house (on a not-so-loaded circuit) for the other device.
The last option might be the simplest. If your washing machine
is on a separate circuit, you might be able to share the fridge's circuit with the dish-washer without any hassles.