I personally like the d-ring happy mouth that you showed, although I don't think this is a biting issue.
I agree with what is said above, that you must keep your leg on even if she's rushing. I would focus on keeping your elbows really relaxed and supple. I like to count strides if my horse is getting rushy, start at one and count however high you need to until you get to the jump. I usually start counting before I make the turn to the jump.
Also, does your saddle fit ? My mares saddle was pinching her and she seemed fine on the flat but was rushing over fences.
I've had the same saddle since I bought her and the rushing only started last year. For 5 years we had the same full cheek snaffle, but it wasn't a good fit and after having three trainers tell me to switch to a better fitting D-ring, I caved. I knew it wasn't a good fit before, but she performed alright in it so I didn't see a need to change. I grew up a bit and started more serious showing and realized it really didn't fit. As soon as I switched to the D-ring last year, I noticed the pulling problems. That's the only reason why I think it might be the bit. Again, I'll be sure to take a lesson and consult a trainer to see what they think is best. The problem isn't the horse, so it's either me or the tack. I don't think she's in any pain, and she got her teeth floated last year.. I'm having a vet do a full check up on her in march since I need my coggins renewed so I will be able to rule everything out before getting a new bit.
LEGS. You must have your legs on her at all times. When she rushes, you probably take your legs off. She thinks "ooo goody. I've gotten her!". Ground poles help a ton. If you can get her to stop rushing at ground poles and get good distances, you can do fences.
Personally I wonder what would happen if you stopped trying to fix all your issues with changing the bit and went back to the basics that worked for you when you rode her bridle less? Obviously you would have to start on the flat, but it may help. I always remember the more they have to brace against (bit and hands) the more they will. Just a thought. Keep it simple.
To be honest, I use two different bits for hunters and jumpers. I mainly use a happy mouth elevator for jumpers but you do have to be careful that if your horse bites into it that you smooth it out (I generally take it to my dad and he sands it down so there are no rough edges because my pony likes to chew on her bits).
And then I use one of these with two reins:
My pony absolutely LOVES this bit. It has the same sort of leverage you get with an elevator, and with the double reins you only apply as much pressure as you need (once you use it a few times you'll get the hang of both reins and how to use them separately of each other). The rubber has some give so that the bit never bangs on the bars of your horses mouth (it also gives your horse something to chew, I know my pony likes to chew).
In all honesty I don't find much of a difference between the two bits, I've been very happy with both. I used the elevator first and then the tom thumb as the elevator substitue in the hunter ring.
Also if your horse is rushing after the jumps, try going a couple strides and then halting straight, or doing the trot jump halt turn trot jump halt exercise I talked about before. That should really help. If she's not sure what to expect after the jump and there's a possibility you will ask for a halt, she'll stop rushing.
I tend not to like the Mullen Mouth type bits only because my horse was "funny" about them. He would get behind the vertical to evade. I personally like the french link Stubben and let me just say that yes, the bit does NOT FLEX BACKWARDS like a regular snaffle. However, it flexes forward and is also a convex shape, so I have found that it rests comfortably in my horses mouth and he is now unable to grab hold of it and take me for a drag like he could with his regular snaffle. Although, it may not be a bit for everyone.
Anyway... bits and all of that aside... can I ask a question here? In the picture of your jumping her bitless and all that, was she still relatively new to the whole jumping thing? Because in the shots you have posted on some of the other threads of you two jumping bigger, and fully tacked, she looks more fit and more keen over the fences. She is, in my opinion, more fit and more brave with her job... and you two just gotta work through the rushing as a result.
I would recommend using a baucher bit I loooove them for jumping, its a good stable bit, but sometimes a horse will rush if he/she loses balance or isn't confident, if that's the case building up confidence is always a good idea, my mare is currently schooling 3" but I do bring her back to the basics pretty often, also I am not a fan of elevator bits, I do use one though, but I only use it for XC my mare tends to get spookish during XC since she's new too it and sometimes throws a fit and if she bolts we're not stopping till she says so, but if you do use the elevator just use it with a soft hand and make sure you have two reins attacked please! One rein should be on the biggest circle and the other should be on one of the lower rings I hope this helps!
Alright, well I rode yesterday and was very conscientious about my riding and what we did. Warming up, I did a lot of transitions, circles and serpentines. I rode in the D ring snaffle and I set up a one stride line starting at 2'. I had two poles perpendicular to the first one to encourage her to keep in the middle and then I had a canter pole after the second one to discourage her from galloping afterward. I went over the line at 2' a few times, making her stop before we got to the wall, then backed a few steps then allowed her to go forward. After a few times of this, I raised the jumps a bit, kept doing this until the second was at 3' and the first was at 2'6". Then, I did some figure 8s by skipping the first jump and taking the second at an angle, then making a loop around and taking the first at an angle. THEN, to test and see if she'd rush, I merged the jumps into a 2'9" square oxer and jumped it. I was afraid she wasn't rushing the line because I put so many things in place to prevent rushing, but she didn't rush it at all! Neither before nor after! I took it a couple times before calling it quits. I was very happy. :)