Thank you Trogdor and rdg20. You raise good points about how opening the apprentice's Arts do, indeed, have bright-line analogs in character. This has all been TREMENDOUSLY helpful.
Thanks to all for your input. I welcome anyone else's comments.
It's important to remember here that the clause in question in Oath of Hermes is as follows: "I concede to Bonisagus the right to take my apprentice if he should find my apprentice valuable to him in his studies."
It, as written, is not a blanket right to take an apprentice away from someone for petty reasons like vengeance or political leverage. The reason for taking the apprentice actually matters.
The Bonisagus in question should probably be asked to justify his exercise of his right on THAT apprentice vs. any other available one (i.e. ones with more training, functioning intellect above 6 years old and apparent virtues).
Note this also blunts the Bonsiagus' ability to steal apprentices 'being abused', though in this case at least it assumes the apprentice has actually had some training. Someone with Int+Magic Theory score of 1 or more is useful in the lab.
None of this applies to the formal offer of apprenticeship, however. Any magus could do that to an unclaimed Gifted child. Bringing them to Tribunal is stupid and telling people she is gifted is doubly stupid, which begs the question how the Bonisagus knew the child was Gifted in the first place. Note that casting spells on a Redcap's child absolutely will constitute scrying of the 'peer into their affairs' sort. People at tribunal are either the servants or family of some magus, so even checking people for the Gift at Tribunal is asking for trouble - they can't claim ignorance like they could just scanning a random crowd of people.
Your reply at tribunal should have been "What apprentice, that's my daughter, have you been scrying upon my family?"
That's a somewhat exotic interpretation. If I were a Bonisagus, I would snort derisively and say that the Mercere saved me the trouble of opening the kid's Arts, and that he can fix an Arcane Connection for me and free up a season for real lab work - if for no other reason than to protect the principle that a Bonisagus can snatch apprentices wherever and whenever he likes.
As for the latter problem, if I test your child for the Gift, I'd argue that I am not peering into your legal affairs, because if "your" child is Gifted, it's not yours, Hermetically speaking - it's a Gifted child, which inherently supersedes any property right a magus has in their family. Obviously, this one is a legal argument that could go either way at Tribunal.
Gaining any information about a magus or his affairs using magic constitutes scrying. Even "Is his daughter Gifted." Nobody is going to be able to argue that a magus' child is not part of their affairs. It doesn't invalidate other magi offering to apprentice the child, but at a tribunal gathering, using Intellego magic is going to pry into somebody's affairs. The fact that the child has the Gift is irrelevant to the fact that the code has been violated.
Note that being Gifted offers exactly zero Hermetic rights. Only those that have sworn the Oath of Hermes have those. The Order considers the offering of apprenticeship a contract with certain responsibilities that the magus must fulfill - the apprentice still has no rights. The magus can kill the apprentice at any time. So yes, the child is the property of the parent in Hermetic terms, at least for the purposes of scrying.
Now it may well be possible to gain the information that the child is Gifted without using magic - like the Redcap bragging about it. If that's the case, the Redcap has no legal defense against the child being apprenticed by anyone. And in fact, someone trying to apprentice the child will still get the child until the scrying charge is heard in tribunal (7 years later).
Don't bring Gifted children to Tribunal. It's that simple, no good can come from it.
Related question- if the magus in question is not bonisagus, but the child was produced with fertility magic, would it be depriving a magus of their magical power?
There's no real precedent on the question*. When you're at Tribunal, talk fast.
I would say no - an unapprenticed child is not part of your magical power even if you made the child - but I could see a strong transitionalist argument for such a claim.
*Canonically, there isn't a precedent for a maga's child going to another magus in the first place - that's a story seed. The law in the matter is pretty much clear, but there's also a fair amount of moral "aw, hell no!" that could work in the parent's favor.
Is there an 'Aww, Hell No!" though? Fosterage and apprenticeship in trades are really common in the middle ages.
Not forcible ones.
A deeper issue here is whether a mage can be treated as a mortal by a mage claiming his rights. It's well established, I believe, that magi can kidnap an apprentice-child at spear-point if they wish, from mortals, so long as no harm befalls other magi. The same applies to a craft master's apprentice or a noble's foster-child, or a child taken by faeries. Does this apply to mage parents?
Signs point to yes.
It's not as simple, however. A magus who takes a Gifted child from anyone but a magus is usually justified in incinerating any opposition (so long as he acts prudently - be careful when blasting nobles to ash) without running afoul of interference or molestation charges. If a magus is the parent, however, then unlike a mortal parent, they have rights unless you take the matter to Wizard's War - you can't slay them or do anything that deprives them of their magical power unless you want to face a countersuit.
I greatly appreciate all the input everyone volunteered in this thread. I incorporated all of it into the many sessions of my ongoing game. If anyone is interested, you can find my log of the campaign here, written in the voice of one of the Bonisagus magi who was present for the whole apprentice appropriate attempt.
The other parent was a gifted Redcap though, which would mean that the child is decended from Mercere on one side of the family line.
I don't think it was made clear that the Gifted Redcap was descended from Mercere ... although it would be very strange for a Gifted mage not of The Blood to be a Redcap or in the House at all. I suppose an orbus might be adopted in as a 'Redcap' under some circumstances.