Sorry from the outset Im writing stuff you may not want to hear but Ive recently had a similar situation.
I use a breville BES200 occasionally and I thought I had pressure major problems with it choking on finer grinds using a hario mini mill and a KYM hand grinder (similar to the zassenhaus, has positive reviews from those whove owned both).
Recently I have tried my new old (;D) Faema ex-commercial grinder and no matter how fine I grind the breville will not choke. The shots just come out more slowly (even at turkish powder) - up to a minute plus for a 30g shot from a double.
My conclusion is that there is too much dust in the hand grinders due either or both that the grinding area is too small or the shaft wiggles too much.
Perhaps you can borrow a different grinder or try a different pre-grind to see if that helps - a good grinder is going to help any machine. Pre-grind would be poor for drinking but may at least be diagnostic. Perhaps the grinder is on its way out?
Finally, what you were asking:
At sub 500 you could get some pretty good gaggia machines on evilbay, recently the classic has been going for sub 200 including shipping.
I have found that the BES200 brews at far too low a temperature (65C straight out of the group head into the cup) for me to consider getting that brand for myself.