Family holiday in Barcelona, Gravel or MTB?

Hello,

I'll be on holiday in Barcelona for the next two weeks.
I'm quite an early bird while the rest of my family likes to sleep in.

I consider taking a bike to discover the countryside and don't know which bike to take.
I have a gravel bike that I also on the road, and a mountain bike.

I saw the moutains around the city and would like to spend most of my bike time in the nature instead of the road.

Do you consider the tracks in the mountains easy enough for a gravel bike, or is a moutain bike needed?

I'm an avid climber and don't mind a bit of difficulty.

Thanks for your insights,

Yoann