What material is that core? If you made the top and bottom plates from ferrous material, the magnetic field will take the shortest path from the edges around the top to the bottom instead of out and into some adjoining metal door-strike.
A magnetic system like you are trying to make forms what's called a "magnetic circuit" (google it). The field forming that circuit is what you want going through your door strike.
To do that, I would suggest winding your flat magnet so the poles are at the left and right of the long thin geometry instead of top and bottom. Your core would be a sort of elongated, flat "U" shape (or a "[" shape). Only use the ferrous material in the core (preferably silicon steel, "electrical steel"). If you absolutely have to cover the windings with metal instead of plastic, use nonmagnetic steel or brass.
The strike should bridge the poles of the "U", and complete the magnetic circuit. With that 272 amp-turns of MMF you have it should be a good strong electromagnet.
A good rule of thumb is to make the cross-sectional area of core bigger than the sectional area of the copper (look at the size of the core relative to the amount of copper in a power transformer). Less than that and the B-field will be dominated by the vector potential rather than the M-curve, it's more efficient for an electromagnet to have it the other way around.
And a hot one if you run it for any length of time embedded in a wooden frame - that's 6 watts of heat that has to get out somehow.