I think you are mistaken. The FE2 does not have a dedicated battery check, the way it is described in the manual is basically something along the lines "if the needle moves, the battey is ok" - thus if the camera doesnt work, you need to change the battery - much the same as Rockwell says.
Regarding battery life, I cannot comment on Rockwell's argument on why the FE2 eats batteries faster, but my experience is that my FE's lasted up to 3 times longer than the FE2 on the same type of battery. The FE2 manual explains that it is using battery power even in off position. I don't know if the FE also uses power in off position, but the battery in the FE certainly lasted longer in my 20 years experience with those cameras.
To test the FE2 battery you just have to use the exposure compensation ring and then the red
-+ LED must shine in the VF if the battery is strong. If the battery is weak/dead the
-+ blinks/doesn't light on. This is where the actual battery tester is installed in the FE2 : a red LED, exactly as it was in the FE.
I don't see how the FE2 could "use battery power even in off position". If it's off, it's off. What happens is : to switch the lightmeter and shutter circuitry on you have to 1/open the winding lever to its 45° resting position 2/half-depress the shutter release button and then the lightmeter will switch on and the black needle move in the VF. There is a 16 seconds tempo so even if you close the winding lever fully, the lightmeter will stay on for a max. time of 16 seconds, true. But, contraringly to the FE, you can live with the winding lever open at its 45° resting position without fearing any battery drain - as soon as you don't depress the shutter release.
Of course if you half-depress the shutter release when the winding lever is fully closed this will not switch the lightmeter on. So the power security is the same as on the FE there.
I have owned three FE and all of them experienced the shutter ceramic-regulated electronics failure eventually.
Now if I was given a FE, even mint, I would certainly not use it as my main camera during a trip or to record some rare and valuable event, because discovering several rolls of film wasted by some erratic shutter speeds isn't fun. They were made in 1977-1982, and now most of them have oxydized and the common failure they suffer from is likely to happen to a large amount of the remaining stock...
OTOH my FE2 never failed... granted to its improved quartz-regulated electronics.
Another bonus of the FE2 vs the FE is that the exposure compensation ring increments at 1/3 f-stops while it increments at 1/2 f-stops on the FE.
The FE was a great camera of its time. Anyway, but for the marginally interesting flipable Ai coupling tab (and its ability to also accept dome-shaped shutter release prongs), it doesn't
really have any advantage over the FE2.