I decided to see just how much almost quadrupling the dissolved oxygen would actually increase oxygen available to the tissues.
I am not sure if your approach to this is along the right lines, although I am not 100% clear on this myself.
I think this sort of calculation (on how much HBOT increases oxygen availability to the tissues) would be a bit complex to perform, and I would not be sure how to do it, because I don't fully understand the dynamics. But I think we can assume that there will be a huge increase in oxygen availability to the tissues under HBOT, as the following explains.
To start with, note that the reason we need hemoglobin is because the 0.3 ml/dl of oxygen that gets dissolved in the blood under normal conditions in the lungs is insufficient to meet the tissue requirements for oxygen. With speed at which blood flows though the blood vessels, that 0.3 ml/dl of dissolved oxygen it carries just cannot meet the constant oxygen demands of the tissues.
So that's where hemoglobin comes in, because it can carry a lot more oxygen, and acts a sort of "oxygen buffer" of the blood. So when blood carrying the normal 0.3 ml/dl of dissolved oxygen flows into the tissues, and those tissues extract this dissolved oxygen from the blood to meet their oxygen needs, the blood is not left without any dissolved oxygen, because the hemoglobin replaces the lost dissolved oxygen that the tissues have taken.
Thus there is constant exchange between the reservoir of dissolved oxygen in the blood, and the (larger) reservoir of oxygen carried by the hemoglobin, with the latter constantly replenishing the former each time the tissues take some of the dissolved oxygen out from the blood.
Now when you increase the partial pressure of oxygen (via HBOT and/or breathing pure oxygen), this increases the amount of oxygen that can be dissolved in the blood, as we saw earlier, from 0.3 ml/dl to typically around 6 ml/dl. This then increases the dissolved oxygen storage capacity of the blood, since now a lot more dissolved oxygen can be stored in each dl (deciliter) of blood.
So this means that as the blood leaves the lungs and flows into the tissues, because it carries so much more dissolved oxygen in HBOT (20 times more than normal), it is less reliant on the hemoglobin reservoir to replenish the dissolved oxygen that is snatched up by the tissues. But I am guessing (but am not sure) that even at these much higher levels of dissolved oxygen, when this dissolved oxygen is snatched up by the tissues, the hemoglobin will immediately replace the oxygen that the tissues have taken out of the blood as per usual, in order to maintain the dissolved oxygen level.
So the actual levels of dissolved oxygen in the blood during HBOT would be a complex interplay between the starting level of 6 ml/dl as the blood leaves the lungs, the amount of dissolved oxygen snatched up by the tissues that the blood flows through, and the speed at which the hemoglobin reservoir can replace this snatched dissolved oxygen. I am not sure how you would calculate that.
But what we need to remember is that increasing the ambient partial pressure of oxygen via HBOT allows each dl of blood to hold up to 20 times more dissolved oxygen. This comes from Henry's law. So under HBOT conditions, assuming that the hemoglobin continues to efficiently do its job of replenishing the dissolved oxygen that is snatched up by the tissues, we might assume that the levels of dissolved oxygen in the blood will remain close to this very elevated level of 20 times higher than normal.
Some of the complex dynamics of how hemoglobin binds to and later releases oxygen in the blood is detailed in
this article; see: "Chapter 2 – Bound Oxygen in the Blood". I don't really understand the intricacies.
What I have read is that with pure oxygen HBOT at 3 atmospheres, the oxygen dissolved in the blood in the lungs is alone capable of supplying all the body's resting oxygen needs, without needing any help from the hemoglobin. †
Thus during HBOT, I would guess that the tissues of the body will "see" up to around 20 times the amount of dissolved oxygen in the blood, which is a huge increase in oxygen availability to the tissues.
But the actual dissolved oxygen figure I think would depend on how easily the hemoglobin can, under hyperbaric conditions, replenish the lost dissolved oxygen snatched up by the tissues, in order to maintain this elevated dissolved oxygen level. I could not find any info on that, and maybe hemoglobin works differently under hyperbaric conditions.
† Note that the body's resting requirements for oxygen are 250 ml of oxygen per minute. In HBOT, when the dissolved oxygen in the blood is 6 ml/dl (= 60 ml per liter), since the heart in the resting state pumps 5 liters of blood every minute, you can see that the dissolved oxygen alone delivered to the body will be 5 x 60 = 300 ml of oxygen per minute, which is more than enough to meet the body's resting needs of 250 ml.