Wild Olive Overwintering

I have two wild olives (Oleaster, not fruiting ones). Temps are gonna be dropping to below 5C next week and I’m taking in my tropicals. Do I take my olives in as well? Here’s an image of onw for reference to the species (it’s been restyled since the photo): Wild Olive Bonsai tree (not in final pot) - Album on Imgur

I leave my Olive out til about 6 to 7 °C.
At night it goes in the heated green house (temp in the greenhouse is 6°C).
During the day when it is sunny and about 6 to 7°C I bring it back out.

I have a small grocery store Olea europaea (fruiting) that I kept indoors last year. I can also keep it in the greenhouse that is designed to not go below 1C (34F). Which do people recommend?

I have determined that rosemary (Salvia rosmarinus) does not like to be kept indoors under my conditions since two have slowly declined and died the following summer. Olive is from a similar climate which is why I am asking about the best way to overwinter.

Olives are native here in Israel and grow naturally in a Mediterranean climate, same with rosemary I think. We have winter rainfall over here and hot dry summers. So maybe take into consideration that the tree may be naturally wanting to take up water and maybe needs warmth to be able to build up a bit. I’m no expert but that is just how they thrive over here.

Yeah, they’re native here too. My concern is that they’re less hardy in a bonsai container and might not take freezing temps at all.

I’m wondering about keeping an olive in my climate. We rarely get down into the 30F’s here. I wondering if I can keep mine outside even if the high is in the 40Fs. Mine is a fruiting olive.