Haven't found anything particularly good, but I've probably gone as far as I'll go.  I've done some analysis trying to predict how much variance we expect from each turtle so that I know how much to overestimate, and for the non-special turtles I'm predicting:
 

Bertrand: 19.0lb

Chartreuse: 26.2lb

Donatello Dontanien: 21.1lb

Espera: 17.3lb

(Flint is already estimated as a gray turtle as 7.3lb)

Gunther: 30.0lb

(Harold is already estimated as a six-segmented clone as 20.4lb)

Irene: 23.7lb

Jacqueline: 20.0lb

I'm rounding these to 0.1lb even though I'm allowed to go more granular, because if the Tyrant weighs to the same precision we do he will also be rounding to 0.1lb, which means we don't gain anything from more precision (estimating 7.25lb gives a payoff exactly halfway between estimating 7.3 and 7.2).

I'll put these estimates in the parent comment for ease of GM extraction.

The one interesting thing I've turned up is that Abnormalities appear to convey a very large amount of variance: each abnormality adds ~1lb of average weight, but actually slightly over 1lb of stdev-weight.  I suspect that abnormalities are adding weight in a highly-random way: my weight estimates for Espera, Irene and Jacqueline (0-abnormality turtles) are relatively low as a result because my confidence was higher, while my estimate for Gunther (6 abnormalities?) has a lot more safety margin built in.

A simple linear regression analysis on the remaining turtles (everything that isn't a Fanged Gray Turtle or a Six-Segmented Harold Clone) gives the following formula:

• 10.56lb base weight if green...
• +2.02lb if grayish-green,
• +5.47lb if greenish-gray,
• +0.359lb/Wrinkle
• +0.142lb/Scar
• +0.598lb/Segment
• +1.000lb/Abnormality

This does a reasonable job of prediction, but has a residual with a fairly-large ~2lb standard deviation.  Our standard-deviation math suggests that this means we should give the Tyrant answers overestimating each turtle by 2.4-2.5lb, and should expect to lose on average ~35gp/turtle to error.

That seems like we might be able to improve on it, but I'm not sure how.  I haven't been able to find any useful interactions yet.  There does seem to be an obvious explanation of all the traits except Abnormalities being driven by some hidden Age variable: old turtles start getting grayish, are wrinklier, have grown more shell segments and accumulated more scars, and are larger.  However, I'm not sure how actionable this is for us.  

The one thing it does look like I can do is adjust the amount of overestimation I do: it does seem that our estimate is less accurate as turtles get older and larger, and so rather than overestimating by 2.44lb for every turtle I should overestimate the larger ones by more and the smaller by less.  That's not going to be a very large improvement, though.  I feel like there ought to be something else to do, but haven't found anything yet.

The Fanged Gray Turtle seems relatively simple, so we look at that first.

The weight of a Fanged Gray Turtle seems well-approximated by (0.425 + 0.4568*#segments) lb.

This leaves behind a residual that looks roughly like a normal distribution with stdev ~0.357lb.  I'm not able to find any interaction of this residual with any other properties of the turtles - scars, mutations, etc. all seem unpredictive for the Fanged Gray Turtle.

Some quick math reveals that the Tyrant's asymmetric payoff distribution encourages us to overestimate a turtle's weight by ~1.22 standard deviations.  Therefore, we're going to bump up all our weight estimates by 0.435lb in order to flatter His Tyranny.  

(We could bump them up a bit further if we thought that reducing the odds of him having an unflattering portrait of us was worth trading off money for.  However, I actually think we can plausibly use that to extract more money: whatever itinerant artist he kidnaps to do that portrait, we can demand that they give us part of their commission in exchange for us being helpful and sitting for the portrait!  Kaching!)

There's only one Fanged Gray Turtle among the Tyrant's pets: Flint, with 14 Shell Segments.  Our best guess of Flint's true weight is 6.8lb, but we're going to overestimate this to 7.3lb in order to optimize our payoff.

1. Are we unusually careful and competent at weighing turtles in a way that the Tyrant is not likely to be?  If he is careless about weighing his turtles, and introduces additional error, that increased variance makes us want to slightly increase how far we overestimate by.
2. What level of granularity are we able to give the Tyrant in our weight estimates?  I think that an estimate of 7.25lb for Flint is slightly higher-payoff than 7.3lb in expectation, but don't know if that's something I'm allowed to give.

When we look at the distributions of variables individually, there's a startling number (5-6k out of 30k) of green turtles with 6 shell segments (the lowest number, never seen otherwise), zero wrinkles, and zero abnormalities, that weigh exactly 20.4lb.  

They do have varying numbers of scars, though, which makes me incline more towards 'some very particular turtle subspecies' and less towards 'one very friendly turtle that figured out that it can get extra attention by wiping off the mark you put on it and coming by again'.

Harold from the King's pets matches this pattern (and thus presumably is one of these strange clone turtles).

Removing those and looking at the rest of the universe:

• The remaining green turtles now resemble the grayish-green and greenish-gray turtles, making me draw the following three species:
  • Fanged Gray Turtles.
  • Six-Segmented Harold Clones.
  • All Other Turtles.
• Most variables are now reasonably smoothly-distributed:
  • Scars and Wrinkles look Poisson-like.
  • Abnormalities peak at 0 and fall off: that might also be a poisson distribution, just from a lower mean, or might be something else.
  • Weights are bimodal (with one peak around 5-6lb for the Fanged Gray Turtles, and one wider peak around 15-25lb for All Other Turtles).

• Abigail: 23.0lb
• Bertrand: 19.0lb
• Chartreuse: 26.2lb
• Donatello Dontanien: 21.1lb
• Espera: 17.3lb
• Flint: 7.3lb
• Gunther: 30.0lb
• Harold: 20.4lb
• Irene: 23.7lb
• Jacqueline: 20.0lb

• It turns out that all fanged turtles are gray, and all gray turtles are fanged.
• This suggests some kind of speciation by color.
• When we break down by color:
  • Grayish-green and greenish-gray turtles show near-identical patterns - I assume those are the same species and you've just categorized them a couple different way:
    • Weight is positively correlated with wrinkles, scars, shell segments and abnormalities.
    • The first three of these are positively correlated with one another, and probably reflect some hidden 'age' variable.  Number of abnormalities is not correlated with the others, and seems to do its own thing.  (Turtles grown larger with age, and also weigh more per extra mutant tentacle they have grown?)
    • Nostril size has no effect.
  • Gray turtles work differently:
    • They show the same pattern of wrinkles, scars and shell segments being positively correlated.
    • However, weight in this case seems to be almost entirely determined by # of shell segments.
    • Perhaps these turtles grow at a more predictable rate, with one shell segment per year that adds a regular amount of weight?
  • And green turtles also work differently:
    • The correlations between wrinkles, scars and shell segments have broken down.
    • Additionally, those variables have only small correlations with weight.
    • The most predictive variable towards weight is the # of abnormalities.
    • Perhaps these are strange mutant ninja turtles of some kind that are perpetually teenage don't have a regular growth lifecycle?
• It looks like these three species of turtle behave differently enough that I'm probably going to end up modelling the three of them all separately (except maybe the what-I'm-assuming-is-age effect that shows up on both gray and mixed-color turtles).
• My planned next step is to try three independent simple regressions and see how predictive they are for each of those three types of turtle.