As weight loss is clearly your main objective planning your post-fast nutrition in the medium and long term is what will give you the sustainable results you desire.
In the short term you will obviously lose some weight during a 7 day fast. Coffee is unlikely to make a measurable difference to weight loss but by all means drink it if it helps you get through the fast.
I’m just an 18/6 noob that very inefficiently lost 20 kilos over 3 years. Lots of mistakes and far from perfect execution hampered my results. The only thing I did get absolutely right is the fact it was sustainable and continues to be so.
In studies coffee helped weightloss by suppressing appetite mainly due to the caffeine. One study also suggested antioxidants were implicated. (I had to write a paper reviewing coffee and tea and weight loss this past summer). You can find the full text articles if you google. Coffee had a greater impact in those who did not regularly consume it.
I personally fast and drink coffee, teas, and zevia (stevia sweetned carbonated water) and if I need salt or to quell a craving ai’ll have a cup of broth (which is only 15 cals). I read that anything over 35 cals breaks a fast but below that is fine. I have successfully dropped weight doing this method, even if it isn’t the mainstream water-fast standard. Works for me.