I just don't think you are going to get a usable product by adding oil to pre-made soap. I don't think you can "fix" that soap that way. You don't know how much superfat is already in there. There is also salt in it, which may cause issues on a re-batch type fix.
If you must try, though, I would do the following:
Turn a crock pot on high to get the crock warm, when the crock gets good and hot, turn it down to low. Grate one bar of soap up using a cheese grater or Salad Shooter. Add about 5 grams of water to start the soap melting. Add about an equal amount of oil. Stir well, until soap is evenly melted and the oil is completely integrated. Then pour into something heat tolerant to mold the soap. Let it sit in the mold long enough to get firm enough not to get dents when you try to unmold it. Then unmold and let it sit out a couple of weeks to get good and hard.
Just in case you are wondering, this is more trouble than learning how to make soap from scratch, so that you control all the ingredients. After all, what may be drying your skin may not be the lack of extra oils. It may be one of the ingredients already in there.
We will be happy to coach you through making soap, including what equipment and ingredients to get, and good videos to watch.
I second what Susie said. I personally wouldn't even try to rebatch that. You would most certainly be better off learning to make your own. Besides, you can make something that is so much better than the ingredients listed in that soap. Your skin would likely thank you.
Or maybe look for a melt and pour that has suitable ingredients? I'm not at all experienced with M&P yet, but from what I've read, there are a wide variety of bases available and it seems a little more friendly to this kind of process?
Maybe someone else can chime on with some advice.
For the record I agree with Susie and shunt2011 but then we're all addicts looking to add to our ranks.
If you are determined to use this particular soap, you'd be better off to use the MCT oil on your damp skin after washing as a moisturizer. Adding oil to soap in any great amount just makes softer soap that doesn't lather well.
Bear in mind it may not be the actual soap in this product that is irritating and drying to your skin. It might be -- I'm not saying it isn't -- but the preservatives, fragrance, and other ingredients in this product may be the irritants. If you make your own simple, basic soap from palm and palm kernel oils, you can ditch everything else. And then you'll know.
I agree w/everyone else, if I had serious skin issues, making my own soap/body products would be so, so worth it. I have just regular dry skin - although v. dry - and using home made soap makes a huge difference. I made melt and pour soaps for a couple of years before switching to mostly CP b/c I was nervous about using lye, but w/good safety measures it is really not a well-based worry.
Also, if you do not want to make CP/HP for what ever reason, I agree with the others who suggested it, MP is much easier to do (literally just melt and pour if you are OK not adding fragrance and color, which would probably be a good idea if you have really sensitive skin), and a good MP base - I use the SFIC ones - is much more skin friendly than the bar that you are using.
SFIC bases do include a surfactant, but a better one that is in that bar, and only one. Additionally, there are a lot of other potential irritants/drying factors in that bar. You can also add a little bit of oil to MP (about 1 TB per lb of base) if you want to try the MCT.
ETA: Crap, I just re-checked the SFIC bases, they use a lot of coconut oil rather than palm/PKO. Although PKO is v. similar to CO (it is not like palm oil, despite the name) so maybe, depending on what the blend of PKO and palm is in that bar, you are actually OK w/coconut oil. I still thing it is worth trying, and buying a lb to test would not be v. expensive given the potential up-side.
There are topical and oral treatments for your condition, and honestly, as someone who's suffered from a severe skin disorder, I think you should be working with your dermatologist for solutions. You can bring up making your own soap and then find a prescription to use in conjunction. Some derms will push for accutane as a last resort. Accutane sounds a lot scarier than what it actually is.
Finding a nice, open dermatologist willing to work *with* me was a huge game changer.