I don’t believe that spirits are the way to go, largely because the alcohol messes with blood sugar levels.
However, (under Atkins, anyway, different low-carb diets have different approaches) a single glass of wine (taking the place of fruit) after induction should be okay. At maintenance, any low carb beer should be good, if it replaces a grain (Michaelob Ultra and Miller Lite being the best ones I’ve noticed for low carb).
At maintenance, you can introduce spirits (stay away from anything with added sugar, especially liqueurs).
Also, rather than Rum and Diet Coke go with Rm and Diet RC/Diet Rite, as Diet Coke is aspartame based, and is frowned upon because aspartame messes with blood sugar levels in a way that sucralose/Splenda do not.
Preferred sweeteners are sucralose and stevia.
FYI, Diet Coke does have a variety that is available with Splenda rather than aspartame. I’ve only seen it in cans, and it is clearly marked on the label with an oval with the word Splenda in it. It’s not bad...
DH & I have been drinking a Skinny Pirate - Diet Coke and Captain Morgan