Making soap without conditioning oils?

Hi all, first post here but have lurked for some time. Thanks to all who have shared info and experiences while traveling the path to a finer shave.

I'm about to embark on my first soap-making adventure, and have read up quite a bit on the forums here and at B&B. Ordered and received small batches of NaOH, KOH, stearic acid, and a couple of pounds of grass-fed tallow from a nice lady in South Dakota via eBay.

I have a decent grasp on the functions of the basic ingredients in most shave soaps, but one thing I haven't come across in my research is the effect of leaving conditioning oils (such as castor oil, avocado oil, etc) out of the recipe completely. I have a thyroid condition and am fairly allergic to most of the polyunsaturated fats, which include pretty much all of the vegetable oils, so I have to go to some effort in my daily life to avoid ingesting them. Absorbing them topically probably isn't as big of a deal, but if possible I'd like to avoid it.

So for the soap chemists here: what do you theorize would happen if I made a small batch of hot-process soap using only tallow, lye, KOH, stearic acid, coconut oil, and perhaps a few drops of essential oil for fragrance? I'm assuming that the soap wouldn't lather/bubble quite as much (or at all?), it may not glide as smoothly, and perhaps I would miss out on some of the moisturizing properties of the conditioning oil (let's say castor oil for example). Do you suppose I'd end up with an end product that's still shave-able, or would I just have a super thick puck of saponified oils with a waxy consistency that won't lather up at all?

The famous MdC is made with just stearic acid, coconut oil and glycerin, and tallow can be considered a conditioning oil so I think you'll be just fine.