Fighting your shadow is the most difficult part of a Bad Moon 100% Black Cat ("Kittycore") run. The shadow must be defeated using combat items, and the black cat has a 50% chance each round of preventing you from using combat items. There are a few strategies that will work:
Strategy 1 is fairly simple; you just power up and charge in, and hope for the best.
Strategy 2 is nearly as simple. You need to be wounded at the time you hit with the massage oil -- specifically, you need to have current HP at least 400 lower than max HP. We'll cover the math in a moment.
To make strategy 3 work, you definitely need to understand the math involved. Hold on, this will only hurt a tiny bit.
First of all, the shadow has 395 HP, precisely. There is no variance; +ML has no effect. You need to inflict 395 hit points of damage on the shadow, by using healing items/spells. Since no in-combat healing spell can do anywhere near this amount, that means you're stuck using combat items.
Combat items and the amount they heal:
So, by the numbers, 4 red pixel potions are a guaranteed win if you can live long enough to throw them all. Throwing 4 poultices or garters has a ~50% chance of doing enough damage, and throwing 5 of them is a guaranteed win.
Next, you should understand the damage the shadow inflicts on you during the fight. The formula is (90 to 100) + (MaxHP/6) rounded up.
Given this, you can calculate how many HP and max HP you should have going into the fight in order to make one massage oil + one RPP a guaranteed kill (for strategy 3), or to make one massage oil a guaranteed kill (for strategy 2).
Since the red pixel potion does a minimum of 100 damage, you need to do at least 295 with the massage oil. This means you have to have at least a maximum HP of 296, so that you can heal yourself from 1 HP to 296 HP. But if you're maximizing HP aggressively, you won't be likely to win initiative, so you're going to take one hit from the shadow before you get to act. This means you will take a variable amount of damage, but the variance is only 11 points.
So, for strategy 3 it's sufficient to have 307 max HP (but better is good!). You should use every +muscle or +HP equipment and buff you can find, within reason. The sash helps enormously if you have a high ascension count.
Let's suppose you have 320 max HP. The shadow will hit you for 320/6 + (90 to 100) or 144 to 154 damage per round. You care about the maximum here, of course. This means that if you want to end up with 1 HP after the shadow hits you, you should begin the fight with 155 HP precisely. In general, you want (MaxHP/6+101) HP or as close to that value as you can manage without going under.
Getting your HP to an exact value is no longer easy. Prior to the Doc Galaktik change in April 2015, it was possible to heal yourself to any HP value of your choice through a special interface in Doc's medicine wagon. Sadly, this capability has been removed.
So, the shadow hits you, and takes you down to 1 HP, or to 11 HP, or somewhere in between. Then you funksling a massage oil (in the top slot -- that's very important) plus a red pixel potion (or poultice, or garter, but RPP is best). The cat will have a 50% chance of blocking your action. If it blocks you, you'll be beaten up, but you do not use up the combat items. The cat stops you from acting; it doesn't steal the items. So, just cure your beaten-up-ness (tiny house or soft green), get back to (MaxHP/6+101) HP, and try again until you win.
The math is identical to that of strategy 3; you just need slightly bigger numbers. The absolute minimum you can get away with is 406 HP (but again, more is better!).
Use damaging and healing to get yourself to (MaxHP/6+101) HP so that you are reduced to somewhere between 1 HP and 11 HP by the shadow's first attack. Attempt to use a scented massage oil. If the cat blocks it, rinse and repeat. If the cat does not block it, you've won -- it will do at least 395 HP, killing the shadow.
Having higher than 406 max HP gives you more leeway in your initial HP. This is helpful since you can't heal with pinpoint accuracy any longer. If you've got 500 max HP, then you want to heal to 185; but let's say you overshoot a bit and end up with 200 HP. The shadow hits you for 174 to 184, so you end up with 16 to 26. Even in the worst case (26), the massage oil heals 474, and the shadow is dead.