Photos and video, if possible, will help, but based on your description and the videos I found of her sire, I would be looking for something that is balanced, flexible, has a good mouth, a good trot, and some size (not necessarily to add size, if you don’t specifically want something taller, but I’d avoid Casall, Diarado, and others that aren’t consistent in foal size and you sometimes see some that are very small). Her sire has a nice canter but looks like his balance is a bit downhill and he looks a bit stiff in the neck and mouth too. If you are looking for a jumper, I’d look for a stallion with blood, but if you’d like one that also has hunter potential and the upper level jumpers isn’t your main goal, I think that you don’t need as much blood.
Do you have any registries in mind? Zangersheide, the North American Studbook, or AES would give you a lot of options and flexibility, and are all full WBFSH members, but if you have a different one in mind you’d like to use already then you’ll want to keep that in mind when picking a stallion. I haven’t used AES but I have used Zangersheide in the past and now use the NAS so I’m happy to answer any questions you have about those two, if you have any.
Some ideas to start with though - just a range of thoughts that may or may not strike your fancy:
Diktator van de Boslandhoeve - he would be available fresh or frozen and seems to give good temperaments and type for both hunters and jumpers. He has a daughter out of a Cumano mare jumping 1.20m and eventing and he is #8 in the USEF rankings for hunter sires this year.
Ut Majeur de Brecey is also available fresh and could work.
Conthargos - crossed nicely on both Balou and Baloubet mares and gives lovely type and rideability. He died in January so I think the supply of frozen is limited and expensive, however, so you might look at some of his sons.
Chagolando PS (Chacco-Blue Conthargos) is in the US and would be worth looking at.
I think Chaccothage Blue is also an interesting young stallion option. He is by Chacco-Blue and out of a Conthargos mare and scored a 10 for willingness and a 9.5 for rideability at his stallion performance test. He does have Baloubet in the 5th generation but I think that is far enough back.
Diacontinus might check your boxes, other than the flat-kneed trot.
I haven’t seen any of his offpsring but you might check out Ulmar Mail, who is available through New Normandy. He is recommended to improve temperament and rideability and jumps and moves well. I’m not sure if he has been tested for FFS, so you would want to test your mare if he hasn’t been.
Gaspahr is available in the US now and has some offspring I really like. He seems to bring you the wonderful Tangelo temperament with a bit more of an uphill canter and size. Maybe not a flat enough trot or a round enough jump for the hunters though.
Ustinov seems to have a number of hunter kids out there.
Etoulon VDL is [EDIT: NOT a hunter, wrong info, sorry!] /now a hunter in the US, so might be interesting./
I think Solid Gold Z might actually be a good hunter stallion with the right mare, although I don’t know about whether this would be the right cross for that. He is a lovely type though.