Hi there! Can I throw like, a whole bunch of training stuff at you? It sounds like excitement / frustration whining, because he can't wait to work? This is entirely fixable with training, but will require you and everyone who cares for this dog to be absolutely stubborn (more stubborn than a jrt!) and entirely on the same team.
Does he whine at home too? Like whining while his food is being prepared, whining when he can't reach a dog on the other side of a fence, etc?
In general, in your day to day life at home, nobody is to give him any attention/play/food/anything if he is whining. Everyone /must/ wait for him to stop whining before giving him anything. At first a second of silence will do, then he'll have to have been quiet for 2 seconds, 3, 4, and so forth, before he'll get any attention of any form.
Note that attention = eye contact, touch, or speech. If you roll your eyes, you sigh, you snap "oh give it a rest!!" at him.. that is attention. For a dog that is super eager to work with you, that is a reward.
What we want him to learn is that whining stops all the fun. Whining makes the human who is preparing his food stop and walk away. Whining makes the human who was about to throw a toy stop and walk away. Whining is no longer to be rewarded in his life in general. You wait a moment and return to making his food/play/whatever, but then walk away the moment he whines again, repeat till you want to pull your hair out.
Whining means you do not have a dog at that moment. You don't hear any dog, what dog? This is vitally important, because if after half an hour of whining you do give him a morsel of attention- he's gonna learn oh hey, I just had to whine long enough and it pays off. That would be intermittent reinforcement at its finest, that's exactly what you'd do if you wanted to train a whining champion.
It would be much better for him to learn this in situations that are not as full-on as working/hunting with you. So if you know he whines when you're getting ready to go walkies, or having his dinner prepared, start there. Don't wait till he's in a situation where his brain is so wired that he can't even think straight, can't even take treats. You can't possibly expect him to stop whining when tied up between feretting, if he can't even be quiet at home.
What happens if you take him away from his dinner bowl halfway through and tie him up a short distance away? if he can't be quiet then, you can't expect him to be quiet when working. So find a "diluted" version of the situation where he whines, and use that to teach him, being quiet and relaxed works. Being quiet and relaxed gets you more attention, more food, more play. He can learn to be quiet just as stubbornly as he can learn to whine.
If he's a smart and stubborn little jrt, he's now gonna come up with creative new ways to get your attention. Do not, for the love of all that is holy, reward any of them or you will create a monster.
Simultaneously, he's gonna need some arousal (no, not that kind) lowering exercises. Sniffing for food on the ground (in grass or on a snuffle mat) is good for that. Put that on cue and use it to reward him for quiet behaviour in future. By put on cue I mean say "sniff the treats" or whatever, and then drop a bunch of small, broken up treats/kibbles that he has to sniff around for. Later, when he's quiet, you can say your word/phrase, and deliver the reward. If he's too excited to eat, then he's too excited to be working.
Also: Build yourself a relaxation command. Dogs can learn to relax themselves, just like they can learn to sit or down. I've heard of dogs going from digging to relaxing just from the owner going "eeeasyy.." in a low tone.
You can build this with association. When he's relaxed (like he's chilling on the sofa next to you / at your feet / whatever) say your future relaxation signal. Something that can be said in a low, stretched out tone, and repeated. Or, your signal could be visual/tactile, too. Like a blanket, or petting his back in slow, calming motions. Later, you can use that when he's a little too excited/frustrated. It wont make him fall asleep, but it'll take him down a notch. It takes a while for this association to build up though, an you have to always recharge it, and you have to work your way up to using it in situations where he's super excited.
I'll admit, I don't have experience with working hunting dogs. But that's the general learning theory answer to your question, to the best of my knowledge. The whining was likely reinforced by him eventually getting what he wanted (jrts are darn smart, so even if you waited for a moment of quiet before unpegging him, he might still have learned the behaviour chain of intermittent whining), so it's likely a learned behaviour, plus excitement/frustration at having to take a pause from working. Impulse control exercises might also be helpful but I feel like I already just drowned yall in my yapping.