Can you post a picture of your "normal western snaffle"? I just want to be sure we are thinking of the same thing, since some bits are incorrectly called a snaffle by some people. Not saying you are one of them, but better to not have any confusion at all about what bit we are talking about.
As far as her not walking through the gate on her own .... you will get different opinions on this, but my opinion is that your horse is NOT finished on gaming events. Therefore she should WALK through that gate if you ask her to. It'll be a little different when you have a finished barrel horse and you allow them to get on the muscle before a run, because you wouldn't want your finished barrrel horse to walk through the gate at a rodeo, because you are there to win some money and you want your horse pumped up and ready to go.
When I am training a young horse on barrels, they will WALK calmly through that gate if I ask them to. That's what I think a horse in training should do, because I don't want them to get into the habit of doing what they want when we are near the gate. I want them listening to me.
So naturally, that's what I think you need to do with her at this point. She's not finished yet and you don't want to create problems at the gate ... especially anxiety problems, when she isn't ready to be run full out yet. I think you need to work with her so she will calmly walk through that gate on your own.
When the ring is open for riding before the start of the show, walk her in and out of that gate, until she realizes there's nothing to get nervous about and she walks on a loose rein.
I personally wouldn't do more than a trot for her first show back next year, and make her walk calmly through that gate and show her there's nothing to get all excited about. So she learns to listen to your cues.
I've gone to a lot of different shows with different rules. Once your horse is finished, you can use those cues for each show, depending on the rules and depending on the show.
Here's some examples. These are all on my "passed on" horse Beau, who was finished on barrels and gaming events. I could walk him in the arena like we were going to do some western pleasure, or he could blast on in there, based on MY CUES to him. He listened. And that's what I want my horses to do.
Sometimes, Beau would get really nervous too and it was best to just walk him in so he didn't get "stuck" mentally in the alley. This was one of those days. So I walked him in. There were no particular rules at this show for how to enter the arena.
Please excuse my hideous pole run ...... We hadn't ran the poles for over a year, and then up and went to a show. But! This particular show had a rule that you could not cross the timeline to start your run until the entry gate was closed. Beau was used to doing running starts in this arena because we had come to some barrel jackpots in the past here. So I let him partially blow on in there, make a circle to give the gate time to close, and then off we went. I could have chose to walk him in there, but I chose not to. He wasn't running as hard on an earlier event when I walked him in so I tried this strategy that worked better to get him "firing" more. (Didn't help my poles though....)
I guess my whole point is: Be able to control your horse at ALL times and enter the area at the pace YOU choose for your horse; not what your horse chooses. So when they are finished, you can make the approach that is going to be best for your horse.
Especially if she won't walk through the gate because she is too nervous, be very careful you aren't pushing her too hard too fast. She may need a slower pace for a while to get the hang of things without getting too stressed out. It's fine for a finished horse to get nervous (after all, I get nervous before competing too!), but I don't like my horses in training to be nervous. I want it to be a relaxing learning experience until I start pushing them and asking for speed.
At the last barrel race I took Red to, he dang near fell asleep before we went in the arena! His exhibition was crummy as a result, but I was over-joyed that he was that calm at a jackpot by himself. (Considering his first jackpot he self-imploded a couple times from nervousness.)