Unlikely but a good number want those black boxes fitted.
If i were you i would go to a car insurance comparison site like compare the market, enter all your criteria and on the quotes it will say "x required" - majority of the cheapest will say, fit black box, they wont mention dashcams.
A dashcam could however be a benefit if you were in a no fault accident and you needed evidence to help prove your case to avoid losing your no claims bonus.
First year insurance an absolute nightmare, my daughters insurance 18 on a mini one was something like £1700 in the first year, practically only a handful of insurers would cover her. She ended up with some crappy insurance company called sabre or something that had terrible reviews.
Good news is however at the renewal with one years no claims we had loads of options available and the price came right down to practically half and she was able to move to Churchill without a black box.
Good luck