FE'ers do claim photos from the moon and from space are false
This part.
The argument is not that all claim images are false, but rather that they're not particularly trustworthy due to their unverifiable nature. Further to that, they are rarely conclusive either way even if we ignored their other problems.
Pictures of the Earth taken from far enough away unequivocally show the Earth to be a globe. And there are a lot. From Apollo missions, from DSCOVR, from geostationary satellites (not only NASA's)...
Someone who claims the Earth is flat also claims these pictures are false. That's a logical equivalence.
By contrapositive, someone who says these pictures might be real also says the Earth might be round.
I'll elaborate a little on this.
Let's take for example the
solar eclipse of December 26, 2019.
It was independently captured by Russia's
Elektro-L No.2 and by Japan's
Himawari 8, both being satellites in a geostationary orbit.
You can find a movie made from the Japanese satellite's images on that day here :
http://himawari8-dl.nict.go.jp/himawari8/movie/720/20191226_pifd.mp4 . You can also view an interactive animation
here (disable shorelines for more clarity).
The Russians don't offer videos or interactive animations, just a FTP with their pictures :
ftp://ftp.ntsomz.ru/ELECTRO_L_2/2019/December/26/ (relevant folders are 0600 through 0800).
We can verify that everything matches : the shape and movement of the clouds, the location of the eclipse. And these pictures are obviously only compatible with a round Earth model, and not compatible at all with any flat Earth model. They are very, very conclusive.
So, there are really only two options.
Option 1, these images are real, and not significantly altered other than for color enhancement. Which, a bit like HDR or color filters on your camera, can change things like contrast and make the picture generally look better, but will not show a cube where there is a sphere or a banana where there is an apple. The Earth is a globe, end of story.
Option 2, these images are fake, or at least have their geometry very significantly altered to match how it would look on a sphere. Then there is an international conspiracy that involves the Russian and Japanese governments to make everything look consistent.
It's anyone's choice to believe in and defend either option. But it's not logically possible to both claim the Earth is flat, and not to claim pictures taken from geostationary satellites are fake.
NB: I chose an eclipse because it's more spectacular and I like eclipses, but you could also check and compare for virtually any other time and date.