Jill - I know you asked Douglas, but I believe it's healthy to exercise both my pancreas and liver. From fairly early on, I was keen to exercise my liver, and once my levels moderated to non-diabetic levels, I have been exercising my pancreas a bit more. Having been a reduced carber, I have always exercised my pancreas, a bit.
For me, once I started testing, and understanding the feedback I was getting from it, I knew I was periodically experiencing liver dump. Whilst it didn't always feel that great, if I went a bit low, and felt really hungry, I didn't go for anything particularly carby to deal with it. I would have a cup of tea, with milk (and therefore some carb), just to nudge me up a bit. It felt it had to make sense to encourage my liver to cycle its glycogen stocks and refresh them every now and then. Doing this, over time, I realised that the liver dumps were less extreme. Initially, I was rising 0.8, and these days it's only 0.3-ish.
Moving forward, I've just applied a similar approach with my pancreas. The science suggests some of our ability to deal with carbs is training, as supported by the need to eat c130g carbs a day prior to a GTT. Low carbers allegedly can do do less well in GTTs, simply because they are not so used to eating carb. It's almost like the pancreas is a naturally lazy organ, and reacts to stimulus about a day after eating. I've put that very badly, but if I have a period where I am lower carb than my usual, for whatever reason, the first time I have carbs after that my post-prandials will be slightly higher than if I repeat the exercise next day. So, it's like my pancreas is caught on the hop first time around, but more ready for it on the second occasion. I don't know if you have seen my type "use it or lose it", but this is where I get that from.
Clearly reintroducing carbs is a personal choice, which for me has been partly driven by the need to stop losing weight, and the prospect of more fat didn't appeal. I tested very diligently as I was doing this, to ensure I wasn't really over stepping the line. I think I am very lucky to be able to eat as I do. I appreciate not everyone can. I'm not as bold are Andrew or Douglas, but I am bolder than many, and during the most recent period of increasing both the quantity and variety of carbs my HbA1c has gone from 34 to 32. Perhaps counter-intuitive, but I'll roll with that particular punch! I must admit I was a little bit nervy this time around, just in case I'd been having significant highs I wasn't recording.
I don't know if that helps you at all, but if you would like to know anything else, just ask.