The question itself sounds fallacious the way OP has phrased it.

This makes sense, you can't invest and save the same unit of money simultaneously.

This doesn't follow from the first clause. The fact that savers don't invest doesn't preclude the existence of investors, and in fact somebody can be both a saver and an investor.

Therefore OPs question:

is seemingly irrelevant. Yes it would be bad if nobody invested in anything, but there's no reason to suspect that investment would cease entirely on a Bitcoin standard.

The reason people tend not to invest their Bitcoin today is because there are no investments that can generate a positive return in Bitcoin. It is always more profitable to simply save the Bitcoin. Once the whole world is on a Bitcoin standard, it is very likely that Bitcoin's value will stabilise, making it possible for shrewd investors to invest it profitably.