Well, it could be the engine or it could be the chassis or it could be the ID plate
Bear in mind the chassis number is stamped on the inner guard, which is a bolt-on part. If the car has ever been in a serious front-end bingle, it may well have had that replaced. Furthermore, the ID plate is only riveted onto the firewall, so they can move too.
Having put that fear of God into you, a licked finger in the air estimate from your ID plate suggests that it would be a late (November?) 1956 build. (My car is early September, #5678. Yours is about as far along again.)
One way to tell if it really is a '56 is to have a feel under the rear wheel arches. If the bodywork there is a single skin of metal, she's definitely an early. If there's a couple of layers, it's a '57 body. There's also a support from the inside of the rear wheel arch up to the rear parcel tray that was introduced in '57 that an early won't have. If you had a later FE or FC to compare to it's a bit easier to identify what changed.
Early FEs also have a body coloured steering column and a few other differences, but if the car has been restored/modified over the years it becomes a bit harder to tell.
Hope that's some help to you.
RET