How can I do this without using a hexagonal cutter? I imagine there must be some geometry hack to accomplish this!
You can cut the cake almost by eye, with some guidance.
Choose a blade like a decorator's knife, with a width equal to the side length you want for each hexagon.
First cut off the top left corner with a red line on the diagram shown. It's the length of the blade you chose, at an angle of 60°. You could gauge it with a geometry set-square, a protractor, or by eye.
Now mark one blade width down from the cut on the left edge, and two blade widths down from the other end on the top edge. The blue line shows the second cut.
Next, make the vertical cut where the green line is. All these cuts are made by pressing the blade vertically into the cake.
Observe that every cut is one blade width and the vertical uncut gaps are two blade widths. This knowledge should allow you to use the knife itself to mark where the next cut will be with a slight ding made with the corner of the blade.
When you have been down the left edge, you should now be able to see how to mark off the next strip. With a little practice, you can probably do this quite quickly, completely by eye.
If you damage the icing unacceptably when the knife is withdrawn, consider cutting the cake before it is iced. Then ice it all at once, later gently separating the hexagonal pieces. If necessary, turn the cut cake over with another board, and use a knife to separate the pieces from the back, and apply the decorative sprinkle last. Note that the raised edges of your baking tray won't impede the cutting at all.
It might be worth making a rough sketch of where the cuts will be, to see whether there is less wastage if the tray is turned by 90°.
You can give the left-over parts from the edges to the dog.
Easiest way... Get someone to 3D print you a 'cutter' that will fit inside the baking tray. Basically just a cookie cutter but rather than doing one cookie at a time it does the entire cake on one press. (Or a few depending on the size of the printed cutter.)
Then just put it on the cake.. press.. profit.
You could try to use some wall-repair tools, hacked to fit your needs. I have in mind something like in the pictures below.
You will need to adjust the angles of the tools. And to do some practice before you use them for important stuff.
Another way is to just use a normal knife, and then develop a lot of skill. I am quite sure it is doable.