I used an FPGA in my original design but it has proven to be complicated and expensive. The Xula LX25 is well documented but produced in low volume and seems to be no longer for sale.
Luckily, Henner Zeller made a polygon laser scanner, the LDgraphy, with a beagle bone black / green. With the LDgraphy firmware a laser scanner can have a clock speed of 2.7 MHz. This is more than sufficient for a transparent polygon scanner. I looked at Zeller's code and it is actually very accessible. With a couple of minor changes it should be suited for the Hexastorm.
I also had the luck of finding an electrical engineer, Salvatore Puglisi, who could design me a board. Result is below see figure 1. I have added the schematics of the board as pdf to the project repository, see firestarter.pdf. I also added the gerber files in  a zip.
I use a simple voltage divider to detect the signal of the photodiode. This worked with the Xula LX25.
Zeller uses a Schmitt-Trigger to have a clean signal for the BeagleBone. I already tested the firestarter board and this is not needed.
Zeller also didn't mount the laser driver on the cape and made his own laser diode driver.
I decided to go for the IC-HKB from IC-haus. It is not seen in the picture and needs to be mounted on the board at U2. Maybe in a future version, I will make one PCB per laser scan head and one central board with stepper drivers.