consider the impedance of the capacitance formed by the dielectric glue and the metalization layers, at microwave frequencies the impedance is fairly low. But it still is a weird discontinuity.
You can make a better horn by buying a copper sheet, learning how to cut copper nicely with good metal cutting scissors, glue it together with temporary super glued on brackets and then solder it together. It's actually going to be easier then making bicone antennas.
A trick would be to use heavier copper at the end launcher sections, but what I see done on old prototypes was that a piece of wave guide was modified. I think the geometry of the end launchers matters a bit more then the tapered part of the horn.
I think the main problem your going to run into with fabrication is the hole drilling, for things like attaching a coaxial connector. So get yourself a step drill (looks like a chirstmas tree), which will cut very nice holes into thin materials. You can also glue or clamp it to a sacrificial material like aluminum bars for drilling, but I really recommend a step drill
The other problem with copper is that it work hardens. A trick for this is to throw the copper, if it gets damaged, into a BBQ or such until it gets dull red hot, then dunk it into cold water, it will get oxidized but come out extremely soft, but oxidized. You can take the grate off a common BBQ grill and lay copper sheet in there until it gets soft, useful if you want to reuse old copper thats been hammered etc (from a roof or something).
If your cut did not come out perfect, since there are no curves used in common horns that are not broadband, you can temporarily glue copper (say with contact spray adhesive) to something metal that is strait and carefully use a file to shape it to the correct dimensions and then free it with solvent. For precise marking, I would recommend you get a flat plane, like glass, glue the copper to it with spray adhesive, sandwich is t down under high pressure (some weights ontop of a another piece of glass with some wood on top), let it glue to the glass plate (which will be very flat), mark it out carefully using a caliper and scribe lines, then unglue it and cut it out. You can get pretty precise stuff this way.
If you want to scribe a line nicely, get yourself a metal workers ruler, which has a flat edge on it at 90 degrees, preferably a thicker one, and a scribe tool that has a flat blade (not one of those round point tip ones that is carbide for general work). Line up the ruler on the marker points and align the scraper so that the flat side is flat against the ruler
If you have trouble imagining the correct scribe tool, imagine a rectangular prism made of tool steel, like the size of a pencil.
Grind it like this : /___________| from the side and kinda like a F117 nighthawk from the front, with a flat nose, so you can drag it on a 45 degree angle with it making firm contact with a significant part of the metal working guide ruler. You will get very consistent results
If your ruler has a good shape, you clamped things down nicely, use a caliper for marking and you work slow, you will get very nice scribe lines. Just remember to work slow and careful (to the point where you wonder if your an idiot). Work in good light. Practice drawing some lines out on scrap so you don't end up with a scratched up peice of copper. Make sure you scribe fairly deep.
The cutting is kind of the hard part here, you want to get the smallest kind of shears that work for your work, don't use bulky imprecise cheap shears for cutting thin copper. Here you will need to experiment (I have a big assortment of various shears).
Then you need a good iron, aggressive flux and careful construction of jigs to setup your thing to be soldered. Practice making minimum solder joints with minimal fillets.
You can make very nice looking things this way.. my main problem was that I was used to working with paper and I tried to rush things. If it makes you feel better, buying a horn like this will cost you several hundreds of dollars, so working carefully to make something is not that much of a hurt to your pride.