On the start of layout set high numbers for Physics world stepping, for example 50 and 50, or even 100,100. You need to experiment with this setting, it makes all physics calculations more accurate, but also affects performance.
Use bigger chain links with much higher density, decrease the number of links. This will make the rope stronger. Say, if I make 100 links with density 2, the rope stretches a little, but doesn't break, unless I start looping it around poles. Unfortunately, it still goes through poles quite easily.
I guess your method of checking the rope overall lenght can work here, but instead of increasing Lilbot density, you need to decrease its velocity. If the rope becomes stretched, find the nearest pole to the Lilbot and adjust Lilbot's velocity, allowing it to move only towards the pole, not away from it. But I don't know the formulas for that, sorry :)
You can also try applying an impulse that will pull the Lilbot back (at rope angle) when the rope becomes stretched.