I did
some work on blood loss previously. The Blood Points in the most complete version of the system (post #37 if you're curious) work out to
roughly 0.1 pints per BP (the relevant post notes 5 BP is roughly 1 unit of blood, but this is erroneous - it should be around 10 BP to the unit, as a unit is roughly 1 pint; I believe I mixed up the "1 unit is approximately 1/2 liter" as "1 unit is approximately 1/2 pint). An HT 10 person recovers BP at a rate of roughly 1 every 4 days, or 0.025 pints per day. Thus, a pint of blood requires roughly 40 days to regenerate. Higher HT can result in a higher recovery rate, of course.
In 40 days, a person makes 4/3 their normal monthly wage. Lacking blood doesn't actually prevent them from working, so I'd say a "fair" cost is probably 25% of what this implies, or 1/3rd monthly wage. Adjust as desired; personally I'd be tempted to go as low as 1/10th monthly wage (around 7.5% of the 4/3 month recovery). Instead of using actual wage, however, using the wage that would be associated with a character's Status level may be more appropriate. Note you could also take an approach like modern plasma "donation" centers, where you simply pay the subject for their time.
The crown princess isn't
just a Status (whatever) character, of course, so a surcharge for that
may be appropriate (I'd say treat as though 1 Wealth level higher for royals, 2 Wealth levels higher for particularly important royals, unless you
have to be royalty to get above a certain Status). If there's something particular about her blood that makes it more useful - if the royals have draconic ancestry, if she's a virgin, etc - a further price modifier may be appropriate.
Do note, of course, that the above doesn't guarantee the market will actually
pay anywhere near that. If the princess' blood is no more useful than a pauper's, the "fair" cost for it would arguably be based on a character with Status -2, and if the alchemist is willing to pay more than that for it, it's simply for the prestige of using royal blood. Of course, if such prestige is important to alchemists (and/or their clientele), the above pricing may well be accurate - it doesn't matter if an elixir made using the blood of a street whore is every bit as potent as one made using the blood of a virgin crown princess with draconic ancestry, if people are willing to pay more for the latter, alchemists will be willing to pay more for the latter's materials.