Connect remotely to the box

Hi,
can't you run an ethernet cable between the two buildings?
although 100m is starting to be a stretch....
you would need to install a wifi bridge with outdoor antennas and connect it to a second access point for your clients.
I recommend this type of product to make the link between the two buildings (you need two with a direct line of sight), you connect the first one via ethernet to your router and the second one via ethernet to an access point or another router.
PS,
be careful, your clients will use your public IP address to surf the web so you will be responsible.
or do you want to broadcast a public access point?
another solution:
an access point behind the bridge

and ... There you go!

Hello,
There are indeed WiFi repeaters (to be logically placed between the two areas to be covered...)
Are you at least receiving any WiFi in your Gîte or not at all?? (test with a laptop or smartphone)
The WiFi standard 802.11N integrated into your box has a range of approximately 100m (keeping in mind that WiFi prefers to pass through windows rather than thick walls...)

"Natural barriers
There are natural obstacles. Depending on the environment surrounding your WiFi box, its signal will be stronger or weaker. All obstacles, whatever they are, absorb part of the energy of the waves. Wood, plaster, synthetic fibers, and glass are light obstacles. Water, brick, and marble present medium interference risks while rolls of paper, concrete, and metal are real barriers that significantly decrease the propagation of WiFi waves."

The problem with radio waves is interference... (Almost all household appliances operate on the same frequencies as Wi-Fi, among others).
In terms of network reliability, I think Orange's solution is the best, even though the cost remains high...

https://www.brack.ch/fr

this wifi signal repeater is small but quite powerful, which extends your wifi range even further. I have it and it covers 250 meters from my window inside. Give it a try.