As I read it, you're setting the crew up for one of maybe three things (there might be others, I suppose):
*overshoot, put the fire out, go back ashore to a runway (you've already said they can't pull off a single-engine landing from 3/4 nm out)
*overshoot, fail to put the fire out, put it in the water, which might be a mite unpleasant as apparently they haven't got the power on one engine to put it on a deck, so putting it into an unknown sea state could be a little problematic
*land on two engines either on the deck or in the water, one of which engines may or may not have been on fire for the last minute or so of flight. Either one of these is going to take you about a minute to pull off.
Or perhaps you're going to be fiendish and immediately make the weather go south on them again as soon as Junior says "I have controls... oh, what's the pretty red light?" so they're forced to overshoot and be indecisive. Hard to say, really.
It's also going to depend upon the actual fire procedure in the aircraft in question. Some older machines have optical or otherwise somewhat unreliable fire detection systems and I've seen a couple which have as their first step "Attempt to confirm presence of fire" or the like. I'll grant you, none of them recently, but still.
Anyway, at 3/4 nm on an ARA, you should already be at Vse (or so close to it that it makes virtually no difference) and level, so you're not going to get a much higher likelihood of success should you choose to shut one down, especially after having burnt almost an hour's worth of fuel.
I don't find the scenario particularly realistic as written (if you get the indication right at 3/4 nm, you've got a full minute to touchdown, maybe 6 steps in a fire procedure and the first (establishing s/e profile) is already done, landing checks should already be done, so it's identify/confirm/do, identify/confirm/do, identify/confirm/do, fire a bottle, whatever...), but then again, maybe I'm missing something. You didn't actually say how many engines were on the machine, either, though I assume it's 2 from post #4.
If there's a 'C', I might attempt to confirm the fire, but again, that's a bit tricky in IMC while as the non-handling pilot you're supposed to be attending to other matters.
What is the instructional objective here? In general, unless you're reconstructing an actual scenario, it's typical to come up with the objective first and then construct the scenario around it.