Then dimming doesn't work
Yes that is correct. The reason is that a filament can not react fast enough to be dimmed by PWM due to the thermal lag. The control region is very narrow.
Other solutions posted unfortunately are rubbish especially the ones that involve torturing small sheep or using diodes. Sorry but there are some people giving poor advice on this forum lately.
What you need to do is to control the voltage to these displays. Perhaps the simplest way is to use the chip as a current sink and connect the common points to +ve through a transistor running as a emitter follower. That is emitter to common points, collector to +ve, resistor on the base to a low pass filter being driven by the PWM.
Very nice PCB and construction by the way.