I've been reading up on Faraday cages in an attempt to answer your question. After rewatching the scene in the episode, I notice that first of all, the exact purpose of the Faraday cage isn't explicitly stated although the obvious conclusion would be to avoid RF detection and protect against electromagnetic pulse.
For reference here are a few images of the Faraday Cage from the episode, including the door that allows them to enter.
So is this a Faraday Cage? Can a Faraday Cage have a door?
I found this article to be a great source for the basics of Faraday cages without a lot of extra science mumbo jumbo.
Can A Faraday Cage Have Holes?
Yes, as long as the holes are small with respect to the wavelength of
the incident electromagnetic wave. For example, a 1 GHz wave has a
wavelength of 0.3 meters in free space. As long as the holes are
significantly smaller than that dimension (i.e., a few millimeters),
they won’t let in much of the incident wave. This is why fine
conductive mesh can be used when constructing a Faraday cage. In
practice, the cage’s lid or door usually causes the most leakage.
Taping the seam with conductive tape helps to reduce this leakage.
So there can be holes but adding a door tends to cause leakage...
Here is another source indicating an almost identical response and example from different site:
Your Faraday cage can have small holes in it, provided they are not
too large with respect to the wavelength of the incoming
electromagnetic wave. This is why you can also use fine aluminum mesh
to build a larger Faraday cage. For example, a 1 GHz wave has a
wavelength of 0.3 meters in space.
Generally with these kinds of mesh cages, the cage door is typically
the part that causes the most leakage, but this can be fixed by taping
the seams with conductive tape.
So in answer to your question, yes this does look like a legitimate Faraday Cage and meets all the characteristics of one, however, you are correct that each time the door is opened they are likely creating leakage leading to them possibly being discovered and they would have been better off to creating a sealed entry and tunnel.
EDIT: I decided to go further with this in regards to the size of the holes in regards to allowing RF frequencies. Here is a chart of the US Frequency Allocations indicating the smallest to largest frequency channels.
Based on that chart I calculated the wavelength size in meters for the smallest (3 kHz) and largest (300 GHz) frequencies using an online calculator.
3 kilohertz = 9993081.933333 wavelength in centimetres
300 gigahertz = 0.09993081933333 wavelength in centimetres
It seems clear that no 3 kHz signals are getting in or out of the cage, however, 300 GHz have an incredibly small wavelength size and its conceivable that these wavelengths could penetrate a mesh cage as we see in the episode. So if the ARV was giving off signals of these frequency then its possible it could have been detected through such a cage.