Note: Cernunnos could easily tame both Lynx in a single season, using his ReAn lab total. In fact, taming one Lynx takes only 10 of his ReAn lab total, which means he could tame both of them and still have about 35 ReAn left for another lab activity that used his Magical Focus, if he chose. This would be done in the lab, and would generate a lab text Cernunnos could use for a (small, +1) benefit when taming more Animal creatures.
Regardless of how the Lynx are tamed, after one season they acquire Loyal (person who tamed them) +0. Additional seasons can either transfer loyalty to someone else or raise Loyalty +1, up to a maximum of +3.
Breeding:
At some point, a "breeding roll" will be made, and you need a 9+ to succeed. Here are your current modifiers:
Competent Handler +1: This is almost certainly going to be Standing Elk, because Gustov will usually have lab work.
Aggressive -3
Carnivore -1
Magical Creature -1: Might 5
Only Beast of its kind besides mate -3
So right now, you would need to roll a 16 to breed the Lynx.
How to improve:
Adequate diet +1: the Lynx are fed, but it's not what they're used to eating or what they prefer. No one has asked them what that is. Of course, they prefer to eat animals native to the Alps.
Agreeable Environment +1: Build them a real lair with space and some kind of natural setting.
Hermetic effects +1 per 2 magnitudes. The fence posts count; I don't know their magnitude. If you could create a cold environment -- with snow! -- they'd love that.
Competent Handler: If Standing Elk's Animal Handling gets to 6, his bonus will go up to +2.
Enlightened Handler: If Standing Elk learns Magic Lore, this will add a bonus of +1-2.
Tamed +3
Similar animals in proximity +1: a long shot, I know. But if you could gather predatory cats of just about any sort, the Lynx would feel less lonely.
So without even Hermetic effects, you can probably raise your chances by +7, bringing the die roll necessary back down to 9. And with magic, you could potentially bring it down to a very manageable number.