Hit points are a poor abstraction for blood loss
Here's the full description of hit points in the Basic Rules, with the most relevant phrases highlighted in bold face:
Hit points represent a combination of physical and mental durability, the will to live, and luck. Creatures with more hit points are more difficult to kill. Those with fewer hit points are more fragile.
A creature's current hit points (usually just called hit points) can be any number from the creature's hit point maximum down to 0. This number changes frequently as a creature takes damage or receives healing.
Whenever a creature takes damage, that damage is subtracted from its hit points. The loss of hit points has no effect on a creature's capabilities until the creature drops to 0 hit points.
The first highlighted phrase indicates that hit points don't just encapsulate physical durability, which makes sense, since things like psychic damage don't necessarily inflict physical harm on a target. More to the point, many damage types, such as fire, cold, lightning, poison, radiant, thunder, and (as mentioned) psychic, may not cause the kind of damage that results in any significant blood loss, despite still inflicting damage on the target.
The second highlighted phrase is potentially important, because it indicates that 0 hit points represents an important threshold: prior to that any injuries or other damage received are not sufficient to impair the creature's ability to act in combat.
With all this in mind, we can come to a conclusion: hit points are a poor abstraction for representing blood loss. Ultimately, the purpose of hit points is to track combat damage, and that doesn't match the situation of a person purposefully letting their own blood. Hence, I would not recommend using loss and gain of hit points to model loss and restoration of blood. Instead, let's look at the symptoms of anemia and try to find a D&D mechanic that comes close to modeling them.
Recommendation: Model anemia with exhaustion
Wikipedia gives the following as common symptoms of anemia (the condition that would result from excess blood loss):
Most commonly, people with anemia report feelings of weakness or fatigue, and sometimes poor concentration. They may also report shortness of breath on exertion. In very severe anemia, the body may compensate for the lack of oxygen-carrying capability of the blood by increasing cardiac output. The patient may have symptoms related to this, such as palpitations, angina (if pre-existing heart disease is present), intermittent claudication of the legs, and symptoms of heart failure.
If we take a look at the consequences of exhaustion in D&D, we'll find a pretty good match for these symptoms:
- Disadvantage on ability checks
- Speed halved
- Disadvantage on attack rolls and saving throws
- Hit point maximum halved
- Speed reduced to 0
Hence, I would recommend that if a character intentionally loses enough blood to cause adverse effects, they should start suffering progressive levels of exhaustion, which are in turn removable by the usual means, such as finishing a long rest and ingesting some food and drink or casting Greater Restoration. (Conveniently, the common means of removing exhaustion just happen to be things that could plausibly heal anemia that is caused by blood loss.)
In addition, I would probably rule that a spell like Regenerate would literally regenerate all your blood cells and thus cure the anemia after some time. This is based on the ability of Regenerate to cure other severe forms of injury not represented by mere loss of hit points, such as those in the Lingering Injuries table of the DMG.
While I originally came up with the idea to use exhaustion to model blood loss from scratch by comparing symptoms to mechanics as described above, it turns out that there is at least one instance in the rules as written where exhaustion is explicitly used to model blood loss: the Living Armor magic item from Eberron: Rising from the Last War, pg. 278:
The armor requires fresh blood be fed to it. Immediately after you finish any long rest, you must either feed half of your remaining Hit Dice to the armor (round up) or take 1 level of exhaustion.
Of note, this item also suggests an alternative means to model blood loss: spending hit dice.