I think you can overcome this.
I believe you will be able to tell when she is going to dive for grass. She'll slow a bit. You'll notice her ears intently focused on a spot off trail. THAT'S when you re-direct her with the reins and give her a good thump with your heels. Throwing in a menacing growl makes the point even stronger.
You mentioned that you wouldn't want to do anything to make her uncomfortable, or something similar to that. You're not. I doubt she is underweight. I'm fairly certain she gets plenty of time off. She's not working very hard. Really. Not at all.
Go for it.
Some friends and I worked at a dude ranch that also had a registered Charlois herd. The horses on the dude string had some lazy habits. But, they are what we used to do all the cattle work with the registered cows, and we competed at open shows on them.
You can do this.
Oh your wonderful advice made me laugh! This is great! I will look for the cues and see if I can't correct her before. I think the one thing I'm getting better at is thinking aloud, making my intent and my body convey to my horse. I think it's because I'm learning to use my body more sensibly and calmly and even assertively, not with reins or anything but I;ve learned not to take no for an answer. It's better to not because then it's a periodic struggle for the rest of the lesson. And I'd rather not frustrate the horse or me.
Yes, those ears. They do tell the tale.
I ride a horse I love named Brave. He's a great boy but he's a lot like this girl Muffin, he will sometimes get lazy and not give me what I ask for. He'll play dumb. He'll play tired. He'll be all "oh...is that what you wanted?...uh...I didn't know" kind of stop/go/slow/fast. I thought it was me not being clearer. But now that I've been assigned a more responsive mare Lily, I know a lot of it was him. So I am looking forward to riding him again. Because I KNOW he's fooling around and being lazy and wanting to eat grass and be turned out and the heck with what anyone wants. If he can get away with it, he will. He's wicked bad! I adore him but he is GETTING FAT being bad.
Horse are a hoot! Whoever thinks horses are dumb is an idiot. THey are smart, cagey creatures and they are quite adept at getting what they want, with either attitude or brute strength. I know they're formidable. I do not underestimate how powerful they are, or how pissed off they can get if you push them wrong. My Lily does not like to halt too much at the beginning of the lesson. Or go backwards. She wants to move for a while. She NEEDS to move, so I let her stretch and move. I'll halt her if I must but I also am mindful that she needs a warmup.
The problem with Muffin is if I shorten the reins, she's so strong, she will GO anyway. If we're on level ground, I can try to maneuver her some. But climbing down sometimes we only have, lets say one horse length between us so I am limited in how much I can do, particularly because there's nowhere to go but down and because she needs her footing.
However, if I can get some level rides with her and show her I mean business, maybe we can establish something. As Tiny said, "read her the come to jesus" riot act, which I might do in my own way. I am not a very experienced rider but I AM getting better.
I don't know what type of bit they use with her but I suspect, being the owner's horse, she has something nice and comfortable and nothing that has the potential to hurt her because they know to not give a new rider that kind of tool. Thinking back that is the impression I have. That she could be steered but not corrected.
Thank you for your advice. You've given me a lot to think about!